LIBRARY 

OF    THE 

UNIVERSITY  OF  CALIFORNIA. 
Class 


FIELD 


OOK 


FOR 


CIVIL     ENGINEERS. 


BY 


DANIEL   C^RHART,  C.E., 

DEAN   AND   PROFKSSOK    OK   CIVIL   ENGINEERING   IN   THE  WESTERN 
UNIVERSITY   OF   PENNSYLVANIA. 


OF  THE 

UNIVERSITY 


BOSTON,  U.S.A.: 
GINN    AND    COMPANY. 

1903. 


COPYRIGHT,  1893,  1903,  BY 
DANIEL  CAJRHART 


ALL  BIGHTS  RESERVED 


- 


PREFACE. 


THE  work  of  the  Civil  Engineer  is  varied  and  extensive. 
He  may  be  called  upon  to  survey  a  tract  of  land  ;  to  lay  out  a 
town  ;  plan  a  system  of  water  supply,  and  sewerage  ;  to  locate 
and  construct  a  road,  canal  or  railroad  ;  to  design  and  erect  a 
bridge,  or  roof  ;  to  build  a  light-house  ;  improve  the  regimen 
of  a  water  course  ;  maintain  a  highway  in  good  condition  ;  in 
fact,  several  of  these  pages  would  be  necessary  to  even  enu- 
merate the  requirements  which  the  modern  engineer  is  ex- 
pected to  meet,  and  almost  every  topic  would  furnish  material 
for  a  manual.  In  this  book  it  is  proposed  to  treat  that  part  of 
the  Civil  Engineer's  work  which  will  enable  him  to  locate  the 
center  line  of  a  railroad  ;  to  set  the  stakes  incident  to  its  con- 
struction ;  to  compute  the  quantities  ;  and  to  solve  the  prob- 
lems pertaining  to  track  laying. 

The  book  is  written  for  students  of  civil  engineering,  and 
to  satisfy  a  demand,  often  expressed  by  field  engineers,  for  a 
manual  convenient  in  size,  containing  the  desired  information, 
systematically  arranged,  fully  illustrated,  and  easy  of  reference 

It  abounds  in  problems,  such  as  are  likely  to  arise  in  ordi- 
nary field  practice,  accompanied  by  full  explanations,  and  with 
illustrative  examples  wherever  it  seems  at  all  necessary. 

The  arrangement  of  the  matter  is  the  natural  one  ;  in  other 
words,  the  various  subjects  are  introduced  and  treated  in  the 
order  in  which  they  present  themselves  in  actual  work.  Ac- 
cordingly reconnoissance,  preliminary  survey  and  location, 
with  the  required  instruments,  occupy  the  first  two  chapters. 
In  Chapters  III.  and  IV.  there  are  numerous  formulas  derived, 


(V  PKEFACE. 

practical  problems  proposed,  and  solutions  indicated  for  them 
in  connection  with  running  simple  and  compound  curves, 
which  are  supplemented  in  Chapter  Y.  by  a  set  of  miscella- 
neous questions,  among  which  several  forms  of  the  Y  problem 
are  discussed. 

Chapter  VI.  treats  of  construction,  consequently  here,  among 
other  things,  are  introduced  methods  of  setting  out  the  work 
and  computing  quantities.  The  difficulties  which  the  young 
engineer  experiences  in  setting  slope  stakes,  were  kept  vividly 
in  mind  when  writing  this  chapter.  It  is  believed  that  the 
subject  is  presented  in  such  a  simple  manner,  and  so  fully 
illustrated,  that  he  can  easily  comprehend  it,  and  make  ready 
use  of  the  methods  explained. 

Chapter  VII.,  on  Frogs  and  Switches,  is  from  the  pen  of  my 
esteemed  friend  and  former  student,  Lewis  C.  Weldin,  C.  E., 
Assistant  Engineer  Pennsylvania  Railroad.  Mr.  Weldin's 
twenty  years'  experience  in  the  engineering  department  of  this 
famous  highway,  qualifies  him  to  present,  with  the  authority 
of  an  expert,  this  subject  in  a  most  practical  manner.  By 
adopting  a  notation  slightly  different  from  that  of  other  writers 
under  this  head,  he  has,  in  many  cases,  obtained  results  much 
simpler  than  any  hitherto  published,  and  he  has  increased  the 
value  of  his  work  by  the  introduction  of  numerous  formulas 
and  examples,  selected  from  his  extensive  practice.  I  there- 
fore desire  to  express  here  my  acknowledgments  to  Mr.  Weldin 
for  this  valuable  chapter. 

The  book  contains  all  the  necessary  tables  for  field  use. 
Among  them  are  included  tables  of  natural  trigonometric  func- 
tions, sines,  cosines,  secants,  cosecants,  tangents,  cotangents, 
versed  sines,  and  exsecants,  as  well  as  tables  of  radii,  long 
chords,  squares,  cubes,  functions  of  a  one-degree  curve,  and 
numerous  others.  The  author  believes  that  time  is  saved  by 


PREFACE.  V 

using,  in  the  field,  the  natural  instead  of  the  logarithmic  trigo- 
nometric functions  ;  accordingly  he  has  omitted  the  long 
tables,  usually  found  in  field  books,  of  logarithmic  sines,  tan- 
gents, etc.  A  table  of  logarithms  of  numbers,  however,  is 
inserted.  The  sines,  cosines,  secants  and  cosecants,  are  on 
tinted  paper,  and  placed  near  the  middle  of  the  set  of  tables. 
All  others  are  on  white  paper  ;  tangents,  versed  sines,  and  ex- 
secants,  near  the  end,  and  logarithms  of  numbers,  radii,  long 
chords,  etc.,  near  the  beginning  of  the  set.  Thus  the  tinted 
pages  indicate  plainly  where  four  important  tables  may  be 
found,  as  well  as  favor  the  eye  while  rea  ling  thereon,  and 
with  a  little  familiarity,  soon  acquired,  as  to  the  relative 
position  of  others,  a  person  can  quickly  turn  to  any  one  which 
he  may  need. 

Besides  the  introduction  of  tinted  paper,  there  will  be 
noticed,  for  the  first  time  in  a  field  book,  the  absence  of 
vertical  lines  in  some  of  the  tables,  and  the  consequent 
facility  and  ease  with  which  these  pages  can  be  consulted, 
will  doubtless  be  remarked. 

In  submitting  this  work  to  the  profession,  the  author  trusts 
that  it  will  prove  helpful  to  the  student  of  civil  engineering  in 
acquiring  a  knowledge  of  field  practice,  and  useful  to  the 
young  practitioner  in  pursuing  his  profession,  as  well  as  a  con- 
venient and  reliable  work  of  reference  to  all  who  use  engineer- 
ing instruments  in  the  field.  With  such  results  accomplished, 

his  aim  will  be  attained. 

D.  C. 

SEPTEMBER,  1893. 


NOTE. 

AT  the  earnest  solicitation  of  those  who  make  use  of  the 
book  both  in  office  and  field  work,  Tables  XX  and  XXI  of  the 
logarithms  of  trigonometric  functions  have  been  added.  These 
tables  were  printed  from  electrotypes  from  Nagle's  "Field 
Manual  for  Railroad  Engineers."  Also,  a  growing  demand 
for  some  method,  easily  applied,  to  pass  from  tangent  to  curve 
or  vice  versa  will  be  met,  it  is  believed,  by  an  Appendix  on  the 
Transition  Curve.  My  acknowledgments  are  due  and  are 
hereby  made  to  W.  C.  Armstrong,  Bridge  Engineer  of  the 
Chicago  and  Northwestern  Railway,  for  permission  to  use 
matter  from  his  publications  for  this  Appendix.  His  method 
will  be  found  extremely  simple  as  regards  both  elucidation 

and  application. 

D.  C. 
JANUARY  3,  1903. 


COISTTEOTS. 

CHAPTER    I. 
RECONNOISSANCE. 

PAGE 

The  Instruments 1 

General  Directions 2,  3 

CHAPTER   II. 

PRELIMINARY   SURVEY. 
A.   THE  TRANSIT  PARTY. 

Adjustment  of  transit 4,5 

The  stadia 6-9 

The  gradienter 10 

The  vernier 11,12 

Duties  of  members  of  the  corps 13-15 

B.    THE  LEVEL  PARTY. 

Adjustment  of  level 15, 16 

Leveling 17,22 

Duties  of  the  leveler  and  the  rodman 23 

C.   THE  TOPOGRAPHIC  PARTY. 

Contouring : 24,  26 

Remarks  on  locating  the  line 27,28 

CHAPTER    III. 

SIMPLE  CURVES. 

A.    DEFINITIONS  AND  FUNDAMENTAL  FORMULAS. 

Definitions , 29,30 

To  find  the  radius  R  in  terms  of  the  degree  of  curve  D 31, 32 

The  deflection  angle  denned 33 

The  long  chord  C  and  external  distance  E  denned 34 


Vlll  CONTENTS. 

PAGE 

To  find  the  length  of  a  curve  L 34 

Observations  and  examples 35 

Remarks  on  field  practice 36 

To  find  the  tangent  T,  given  E,  or  D,  and  central  angle  a 37 

To  find  R,  given  Tand  a 37 

To  find  C,  given  R  and  a 38 

To  find  the  mid-ordinate  M,  given  R  and  a 38 

To  find  M,  given  E  and  a 38 

To  find  Jf,  given  R  and  C 38 

To  find  any  ordinate,  given  R,  C  and  d 39,  40 

To  find  E,  given  R  and  a. 40 

To  find  E,  given  T  and  a 41 

To  find  T,  given  E  and  a 41 

To  find  E,  given  M  and  a 41 

Remarks  on  the  application  of  formulas 41 

Application  of  Tables 42, 43 

Formulas  grouped  for  convenience 43,44 

B.    LOCATING  SIMPLE  CURVES. 

To  locate  a  curve,  given  D 44 

To  find  direction  of  tangent  at  a  given  point 45 

Remarks  on  methods  of  procedure 46 

Methods  of  keeping  notes 47 

Remarks  on  the  notes 48 

To  locate  a  curve  by  offsets  from  tangent 48 

To  locate  a  curve  by  offsets  from  chords  produced 49 

C.    OBSTACLES. 

To  pass  an  obstacle  on  a  curve 51 

To  locate  a  curve  when  the  P.I.  is  inaccessible 52 

To  locate  a  curve  when  the  P.C.  is  inaccessible 52 

To  locate  a  curve  when  the  P.C.  and  P.I.  are  inaccessible 53 

To  pass  from  curve  to  tangent  when  the  P.T.  is  inaccessible.  54, 55 
To  extend  a  curve  across  a  stream , 56 

D.    PROBLEMS  IN  CHANGE  OF  LOCATION. 

To  find  the  change  in  R  and  E  for  a  given  change  in  T. 56, 57 

To  find  the  change  in  R  and  T  for  a  given  change  in  E 58 

To  find  the  change  in  T  and  E  for  a  given  change  in  R 58 


CONTENTS.  IX 

PAGE 

To  find  new  R  for  curve  to  connect  P.C.  and  a  parallel  tangent  59 

To  find  new  P.C.  to  connect  by  same  R  with  parallel  tangent  60 
To  find  new  R  and  P.C.  for  curve  ending  in  parallel  tangent 

at  a  point  on  same  radial  line 62 

To  find  new  P.C.  for  new  tangent  from  same  P.T 02 

To  find  new  R  and  P.C.  for  new  tangent  from  same  P.T 63 

CHAPTER   IV. 

COMPOUND  CURVES. 

A.    PROBLEMS  IN  LOCATION. 

To  find  JR',  given  R,  T,  T'  and  a 64 

To  find  R',  given  JJ,  d,  and  angles  between  the  chord  and  tan- 
gents   65 

To  find  T,  T',  d,  and  the  angles  between  them,  given  JJ,  jR', 

a'  and  a" 66 

To  find  R  and  R'  for  tangent  parallel  to  d,  given  d  and  the 
angles  between  it  and  the  tangents 67 

B.    OBSTACLES. 

To  locate  a  compound  curve  when  the  P.C.C.  is  inaccessible....     68 
Various  conditions  and  methods 68,  69 

C.   PROBLEMS  IN  CHANGE  OF  LOCATION-^ 

To  find  the  change  in  P.C.C.  for  parallel  tangent  outside  the 

terminal  and  last  R  the  longer 70 

To  find  the  change  in  P.  C.  C.  for  parallel  tangent  inside  the 

terminal  and  last  R  the  longer 72 

To  find  the  change  in  P.C.C.  for  parallel  tangent  outside  the 

terminal  and  last  R  the  shorter 73 

To  find  the  change  in  P.C.C.  for  parallel  tangent  inside  the 

terminal  and  last  R  the  shorter 74 

To  find  a  P.C.C.  from  which  a  curve  of  known  R  may  run 

and  end  in  a  parallel  tangent 75 

To  find  a  P.C.C.  and  last  R  for  curve  ending  in  parallel  tan- 
gent at  a  point  on  same  radial  line.  Four  cases 76-79 

To  find  P.  C.  C.  and  change  last  R  to  end  curve  at  some  other 
point  in  terminal  tangent.  Two  cases  79-81 

To  substitute  a  three-centered  compound  curve  for  a  simple 

curve .  81 


X  CONTENTS. 

CHAPTER   V. 

MISCELLANEOUS  PROBLEMS. 

PAGE 

Given  a  simple  curve  intersected  by  a  line,  to  find  a  point  on 
the  curve  whence  to  run  a  curve  of  given  R  to  end  in  the 

line  as  tangent.     Two  cases 83,  84 

Given  a  tangent  and  a  curve  to  connect  by  another  curve 

forming  a  Y 84 

Given  a  curve,  and  radii  of  two  others,  to  connect,  forming  a  Y. 

Two  cases 85,86 

To  lay  out  a  track  of  given  length  having  circular  ends 87 

To  substitute  a  simple  curve  for  a  tangent  between  two  curves,     88 

To  locate  a  tangent  to  a  given  curve  from  a  point  without 89 

To  locate  a  definite  point  in  a  curve  from  some  point  in  the 

tangent....; 90 

To  run  a  curve,  with  a  given  .R,  from  a  tangent  which  shall 

pass  through  a  given  point 90 

To  prolong  a  line  beyond  an  obstacle.     Several  methods 91,  92 

To  find  the  radius  of  a  railroad  track 93 

To  locate  a  curve  parallel  to  a  given  curve 93 

To  connect  two  parallel  tangents  by  a  reversed  curve.     Two 

94 


CHAPTEE   VI. 

CONSTRUCTION. 
A.    GENERAL  DIRECTIONS,  DEFINITIONS  AND  PROBLEMS. 

Definitions 96,97 

To  find  where  two  grades  will  meet 97 

To  find  where  a  grade  will  pass  from  cut  to  fill  and  vice  versa    98 

Vertical  curves 99 

Difference  in  elevation  of  the  rails  on  curves 100-102 

B.    SETTING  SLOPE  STAKES. 

When  the  ground  is  level  transversely.     Formula 103 

When  the  ground  slopes  transversely.     Formula 104, 105 

Side-hill  work 106, 107 

Compound  section 108 

The  common  practice  of  setting  slope  stakes 108-110 

Shrinkage Ill 


CONTENTS.  XI 

C.    CALCULATING  THE  EARTH  WORK. 

PAGE 

Volume  of  prism,  pyramid,  and  wedge 112 

The  prismoidal  formula 113 

Sectional  areas 114,  115 

The  volume  by  prismoidal  formula,  and  by  averaging  end 

areas 116-118 

Excavation  on  curves 118-122 

Overhaul  123 

D.    CULVERTS,  BRIDGES  AND  TUNNELS. 

To  stake  out  culverts 124 

To  stake  out  bridges  and  trestles 125-127 

Tunnels 127-132 

Ballast  stakes 133 

CHAPTER   VII. 
FROGS'  AND   SWITCHES. 

The  gauge  g^  frog  angle  F,  lead  L,  point  of  switch,  etc. ,  defined 

134-136 

Turnout  from  straight  track. 

To  find  L  and  R,  given  F  and  g 136 

To  find  L  and  F,  given  R  and  g 137 

Turnout  from  outside  of  curved  track. 

To  find  L  and  R',  given  R  and  F 138,  139 

To  find  L  and  F,  given  R  and  R' 140 

Turnout  from  inside  of  curved  track. 

To  find  L  and  R',  given  J^and  R 141,  142 

To  find  F  and  L,  given  R  and  R' 143 

Crossovers. 

Between  straight  -tracks 144 

Between  curved  tracks 145,  146 

Crossing  frogs. 

In  straight  tracks 147 

In  curved  tracks ....147-152 


Xll  CONTENTS. 

Crossing  slips. 

PAGE 

Between  straight  tracks 152, 153 

Between  curved  tracks 153-157 

Trigonometric  formulas 158 

Solution  of  right  and  oblique  triangles 159 

General  formulas 100,  161 

Miscellaneous  formulas , 162 

TABLP:S. 

I.  Kadii  of  curves 165 

II.  Tangents  and  externals  to  a  1°  curve 168 

III.  Tangential  offset  at  100  feet 172 

IV.  Mid-ordinates  to  100-foot  chords 172 

V.  Long  chords 173 

VI.  Mid-ordinates  to  long  chords 175 

VII.  Minutes  in  decimals  of  a  degree 177 

VIII.  Squares,  cubes,  square  and  cube  roots 178 

IX.  Logarithms  of  numbers 195 

X.  Natural  sines  and  cosines r 213 

XL  Natural  secants  and  cosecants 222 

XII.  Natural  tangents  and  cotangents 235 

XIII.  Natural  versines  and  exsecants 247 

XIV.  Cubic  yards  per  100  feet,  in  terms  of  center-height ....  270 
XV.  Cubic  yards  per  100  feet,  in  terms  of  sectional  area..  276 

XVI.  Mutual  conversion  of  feet  and  inches  into  meters 

and  centimeters 280 

XVII.     Mutual  conversion  of  miles  and  kilometers 281 

XVIII.     Length  of  V  arc  of  latitude  and  longitude 281 

XIX.     Stadia  measurements 282 

XX.     Logarithmic  sines  and  cosines 290 

XXI.     Logarithmic  tangents  and  cotangents 305 


CONTENTS  OF  APPENDIX. 


PAGE 

Transition  curve,  or  spiral 321 

Angle  turned  by  the  spiral  curve 321 

Properties  of  the  spiral..: ...  322 

Comparison  of  the  transition  curve  with  the  cubic  parabola....  322 
Deflection  from  any  point  on  curve  to  any  other  point  on 

curve 323 

General  rule  regarding  deflections  from  point  to  point  on 

curve : 324 

Formulas  for  semi-tangent  T  and  external  secant  E 324 

Diagram  showing  parts  in  true  proportion 325 

How  to  lay  out  the  curve 326 

Diagram  and  illustrative  example 327 

Modification  of  above  example 328 

Additional  examples 329-330 

Table  I.  Elements  of  a  No.  1  spiral 331 

Table  II.  Deflections  of  a  No.  1  spiral 332-333 


xiii 


ERRATA. 


PAGE 

57.    In  first  part  of  example  at  bottom,  cot  17°  was  used  instead 
of  16°,  causing  an  error  of  one  minute  in  degree  of  curve. 
65.    Tenth  line  from  bottom,  for  o^Fread  of. 

68.  The  letter  A  is  wanting  in  the  figure  at  end  of  line  DP. 

69.  Fifth  line  from  top,  E  should  be  E\. 
74.    In  the  figure  the  upper  T  should  be  2". 

81.  In  eighth  line  from  bottom,  for  radii  read  radius. 

81.  a  in  ninth  line  from  bottom  should  be  a'. 

83.  In  last  line,  for  PD  read  PP'. 

85.  In  Fig.  49  the  0  at  the  right-hand  angle  should  be  (X. 

93.  Right-hand  member  of  eq.  (90)  should  read  2  B'  sin  $  EOD. 

94.  Second  line  from  bottom,  the  first  DT  should  be  TT'. 

cc" 

95.  First  equation  should  read  E'  =  •:—=•  * 

96.  Eleventh  line  from  bottom,  for  affected  read  effected. 
98.    Seventeenth  line  from  top  should  have  —  before  -J-  x. 

12  12 

116.    Third  line  from  bottom,  for  —  read  —  • 

119.  Fourth  line  from  bottom,  for  POP  read  POP'. 

120.  Middle  term  in  tenth  line  from  top  should  read 


120.  Fifth  line  from  bottom,  bGE  should  be  CGE. 

121.  Second  line  from  top,  for  r°¥  read  -^. 
121.  Third  line  from  top,  for  T°ff  read  -1/. 
121.  Fourth  line  from  top,  for  T°2  read  -y-. 

121.  Sixth  line  from  top  should   read,  between  equality  signs, 

J7  (746  +  2180  +  370)  ±<M>. 

144.  Third  line  from  bottom,  for  sin  F  read  sin  F'. 

147.  Eleventh  line  from  bottom,  for  C  read  B. 

148.  First  member  of  eq.  (145)  should  be  ED. 

152.  In  Fig.  93  there  should  be  a  B  at  intersection  of  lines  KD 

and  AE. 

153.  In  eqs.  (160)  and  (161),  for  360  read  180. 

155.  In  eqs.  (165)  and  (166),  for  360  read  180. 

156.  In  eqs.  (171)  and  (172),  for  360  read  180. 

157.  In  eq.  (173),  for  O'O  read  O"0. 

216.  Opposite  the  cosine  18°  57',  for  93580  read  94580. 


FIELD     BOOK. 

CHAPTER    I. 

BECONNOISSANCE. 

1.  In  determining  the  best  location  for   any  highway,  es- 
pecially if  it  is  to  be  of  considerable  length,  it  is  customary 
for  the  engineer  to  make  a  hasty  examination  of  the  country 
lying  in  the  general  direction  of  the  proposed  route  ;  gathering 
facts   regarding   streams,   mountains,  valleys,    soil,    etc.,    and 
other  data  bearing  upon  the  construction  of  the  highway,  the 
business  it  may  command,  and  the  cost  of  operating  it,  seeking 
thereby  to  limit  and  minimize  the  detailed  work  which  follows. 
Such  examination  is  called  a  Reconnaissance. 

2.  The  Instruments  which  may  be  used  advantageously  on 
such  surveys,  are  the  pocket  compass,  locke  level,  tape  measure, 
aneroid  barometer,  and  field-glass. 

a.  The  Pocket  Compass  may  be  used  for  roughly  deter- 
mining the  direction  of  any  line. 

b.  The  Locke  Level  for  obtaining  approximately  the  differ- 
ence in  height  of  two  points  by  direct  measurement,  where  the 
points  are  not  far  apart,  either  vertically  or  horizontally. 

c.  The  Aneroid  Barometer  may  be  used  to  determine  quite 
closely  the  difference  in  elevation  of  two  places  whose  distance 
in  any  direction  may  be  considerable,  in  the  following  manner  : 

Take  the  barometric  reading  at  each  station  ;  denote  the 
reading  at  the  lower  and  upper  station  respectively  by  b  and 
&',  and  the  required  difference  in  feet  by  d. 

Then  d  =  60300  (log  b  —  log  I'}. 


RECONNOISSANCE. 


EXAMPLE. 

If  the  reading  of  the  barometer  at  the  foot  of  a  mountain 
be  28.5  inches,  and  at  the  top,  26  inches,  the  height  of  the 
mountain  will  be 

d  =  60300  (log  28.5  —  log  26)  =  2404  feet. 

3.  In   preparing  to   make   a    reconnoissance,    the    engineer 
should  also  provide  himself,  if  he  can,  with  a  good  map  of  the 
locality  to  be  traversed,  and,  as  far  as  possible,  obtain  from 
persons  acquainted  with  the   country  information  pertaining 
to  the  case  in  hand.     From  a  map  he  will  perceive  the  direc- 
tion,  length,   and   location   of    the   water   courses   and   their 
tributaries,  the  position  of  mountains,  valleys,  etc.     He  will 
ascertain    upon   inquiry   something    concerning    "  high "    and 
"  low "   water,    the   behavior   of    streams    during    floods,    ice 
gorges,  maximum  snow  fall,  etc. 

4.  Ordinarily  in  making  a  railroad  along  a  river,  bridges 
have  to  be  built  to  span  tributaries.     If  the  divide  is  kept, 

few,  if  any,  bridges  will 
be  necessary,  as  may  be 
seen  in  the  figure  oppo- 
site, where  AB  represents 
the  line  near  bank  of 
stream,  and  A'B'  that 
on  the  dividing  ridge 
between  two  valleys  or 
streams.  To  pass  from 
the  valley  of  one  water- 
course to  another  it  is 
sometimes  practicable  to 
do  so  with  easy  grades, 
where  their  sources  are  quite  near  together,  by  following  up 
one  and  crossing  the  ridge  by  cut  or  tunnel,  gaining  the  other 
as  CED.  To  go  directly  across  from  C  to  D  would  generally 
involve  considerable  earth  work,  or  heavier  grades,  though 
there  would  be  a  saving  in  distance.  The  items  of  bridges, 


GENERAL    DIRECTIONS.  6 

however,  on  tributaries,  grades,  cost  of  earth  work,  and  busi- 
ness to  be  acquired  on  the  different  lines,  as  well  as  the  main- 
tenance of  the  permanent  way  and  other  operating  expenses, 
must  have  great  weight  in  deciding  the  route. 


FIG.  2. 


5.  Having  studied  carefully  the  maps,  and  obtained  all 
information  possible  from  other  available  sources,  the  engineer 
proceeds  to  traverse  the  country  in  both  directions,  observing 
its  topography,  nature  of  the  soil,  banks,  beds,  and  accessibility 
of  rivers;  and  objective  points,  such  as  passes  in  mountains  to 
be  crossed,  and  depressions  in  ridges  through  which  the  best 
grade  may  be  secured,  adding  to  his  knowledge  already 
acquired,  and  thereby  qualifying  himself  for  an  intelligent 
decision  in  the  matter  of  location. 

The  young  engineer  will  gain  much  general  information 
bearing  on  this  subject  by  careful  examination  of  some  exist- 
ing lines  traversing  our  country,  especially  will  he  be  benefited 
by  a  study  of  the  history  and  location  of  our  trans-continental 
lines. 


CHAPTER    II. 

PRELIMINARY    SURVEY. 

6.  A  Preliminary  Survey  is  an  examination  in  detail  of  the 
belt  of  country  somewhere  in  which  the  location  of  the  line  is 
likely  to  be  made;  the  data  for  determining  its  limits  having 
been  obtained  on  the  reconnoissance.     The  field  corps  required 
to  make  it  may  be  organized  into  three  parties  as  follows :  — 

THE  TRANSIT  PARTY. 

THE  LEVEL  PARTY. 

THE  TOPOGRAPHIC  PARTY. 

A.    THE    TRANSIT   PARTY. 

7.  The  Transit  Party  is  composed  of  one  chief  engineer  or 
senior  assistant,  a  transitman,  two  chainmen,  two  or  more  axe- 
men, a  stakeman,  and  a  flagman. 

The  instruments  needed  by  this  party  are  the  transit,  two 
sight-poles,  a  hundred-foot  chain  or  tape,  and  axes  for  the 
axemen. 

THE  TRANSIT. 

The  principal  adjustments  are 

1.  The  Levels. 

2.  The  Line  of  Collimation. 

3.  The  Standards. 

8.  To  adjust  the  levels,  that  is,  to  make  the  level-tubes  parallel 
to  the  vernier  plate,  or  perpendicular  to  the  axis  of  the  instrument. 
—  Set  up  the  instrument  firmly,  and  by  the  leveling  screws 

bring  each  bubble  to  the  center  of  its  run.  Then  turn  the 
plate  half-way  round.  The  bubbles  should  remain  centered;  if 
they  do  not,  then  with  the  adjusting  pin  turn  the  small  screws 
at  the  end  of  the  level  until  the  bubbles  are  moved  over  half 
the  error.  Then  bring  the  bubbles  again  to  the  center  by  the 
leveling  screws,  and  repeat  the  operation,  if  necessary,  until 
the  bubbles  will  remain  centered  during  a  complete  revolution 
of  the  plate. 


ADJUSTMENT    OF    TRANSIT.  5 

9.  To  adjust  the  line  of  collimation,  or,  to  make  the  line  of 
collimation  perpendicular  to  the  horizontal  axis  of  the  telescope.  — 
Set  up  the  instrument  on  tolerably  level  ground,  level  it,  and 
bring  the  intersection  of  the  cross-hairs  on  a  definite  point  two 
or   three    hundred    feet    away.     Clamp    the    plate   to  prevent 
horizontal    motion,  plunge  the    telescope,  and    locate  a  point 
now  covered  by  the  intersection  of  the  cross-hairs,  opposite  the 
first  and  at  about  the  same  distance  from  the  instrument.     Now 
unclamp  the  limb,  revolve  it  horizontally  half-way  round,  and 
set  the  intersection  of  the  cross-hairs  on  the  point  first  observed. 
Clamp  as  before,  and  plunge  the  telescope  again ;  the  intersec- 
tion of  the  cross-hairs  should  cover  the  second  point  set;  if  it 
do  not,  then  with  the  adjusting  screws  at  the  side  of  the  tele- 
scope, screwing  one  in  and  the  other  out  simultaneously,  move 
the  vertical  hair  until  it  covers  a  point  one-fourth  the  distance 
between  the  last  two.     Repeat  the  operation  to  check  the  work. 

10.  To  adjust  the  standards,   that  is,  to  make  the  horizontal 
axis  of  the  telescope  parallel  to  the  vernier  plate;,  or  the  locus  of  the 
line  of  sight  a  vertical  plane. 

Set  up  the  instrument  firmly  and  level  as  before,  set  the  in- 
tersection of  the  cross-hairs  on  some  high,  well-defined  point, 
as  a  church  spire,  or  some  projection  of  a  high  chimney.  Clamp 
the  plates,  to  prevent  horizontal  motion,  depress  the  object  end 
of  the  telescope,  and  direct  the  intersection  of  the  cross-hairs 
to  a  point  on  the  ground  a  few  feet  from  the  instrument.  Now 
unclamp  and  turn  the  instrument  horizontally  half-way  round, 
sight  the  first  point,  clamp  as  before,  and  note  if  in  depressing 
the  telescope  the  intersection  of  the  cross-hairs  covers  the 
lower  point.  If  it  do  not,  then  with  the  adjusting  screws  in 
one  of  the  standards,,  raise  or  lower  that  end  of  the  horizontal 
axis  until  the  motion  of  the  line  of  sight  in  a  vertical  plane  is 
assured.  Check  as  before. 


PRELIMINARY    SURVEY. 


AUXILIARIES. 

1°.    The  Stadia. 

11.  The  Stadia  is  a  compound  cross-wire  ring  or  diaphragm, 
shown  in  Fig.  3,  having  three  horizontal  wires,  of  which 
the  middle  one  is  cemented  to  the  ring  as  usual,  while  the 
others,  lb  and  cc,  are  fastened  to  small  slides,  held  apart  by  a 
slender  brass  spring  hoop,  and  actuated  by  independent  screws 
dd,  by  which  the  distance  between  the  two  movable  wires  can 
be  adjusted  to  include  a  given  space;  as,  1  foot  on  a  rod  100 
feet  distant.  These  wires  will  in  the  same  manner  include 
2  feet  on  a  rod  200  feet  distant,  or  half  a  foot  at  a  distance  of 
50  feet,  and  so  on  in  the  same  proportion;  thus  furnishing  a 
means  of  measuring  distances  —  especially  over  broken  ground 
—  much  more  easily,  and  even  more  accurately,  than  with  a 
tape  or  chain. 


FIG.  3. 

12.    Its  principles  may  be  explained  more  fully  as  follows: 

Let  Fig.  3a  represent  a  section  of  a  common  telescope  with 
but  two  lenses,  between  which  the  diaphragm  with  the  stadia 
wire  is  placed,  and  assume  that 

f  =the  focal  distance  of  the  object  glass; 

p  =  the  distance  of  the  stadia  wires  a  and  b  from  each  other; 

d  =  the  horizontal  distance  of  the  object  glass  to  the  stadia; 


THE    STADIA. 


a  =  the  stadia  reading'  (BA ); 

D  =  the  horizontal  distance  from  middle  of  instrument  to 
stadia. 

The  Telescope  is  leveled  and  sighted 
to  a  leveling  or  stadia  rod,  which  is 
held  vertically,  hence  at  right  angles 
with  the  line  of  sight.  According  to 
a  principle  of  optics,  rays  parallel  to 
the  axis  of  the  lens  meet,  after  being 
refracted,  in  the  focus  of  the  lens. 
Suppose  the  two  stadia  wires  are  the 
sources  of  those  rays,  wre  have,  from 
the  similarity  of  the  two  triangles 
a'lt'F  and  FAB,  the  proportion 
d-f:a=f:p. 

The  quotient  / :  p  is,  or  at  least  can 
be  made,  constant,  and  may  be  desig- 
nated by  k,  hence  we  may  write 
d—f=FC  =  ka. 

To  get  the  distance  from  the  center 
JV  of  the  instrument  there  must  be 
added  to  FC  the  value 

c  =  OF  +  ON. 

ON  is  mostly  equal  to  half  the  focal 
length  of  the  object  glass  ;  hence, 

c  =  1.5/. 

Therefore    the    formula   for   the   dis- 
tance of  the  stadia  from  the  center 
of  the  instrument,  when  that  stadia  is  at  right  angles  to  the 
level  line  of  sight,  is 

D  =  ka  +  c.  (I) 

13.  When  the  line  of  sight  is  not  level  it  is  impracticable, 
especially  in  long  distances,  to  hold  the  rod  in  a  vertical  plane, 
and  at  the  same  time  perpendicular  to  the  line  of  sight ;  hence 
it  is  customary  to  hold  the  rod  vertical,  as  in  the  preceding 
case,  and  obtain  the  true  distance  by  applying  a  correction 
depending  upon  the  angle  of  inclination  of  the  sight. 


PRELIMINARY    SURVEY. 


Mt 


This  correction  is  deduced  as  follows  :  — 
Let  AGB  —  2m; 

n  =  the  angle  of  inclination  ; 

CD  must  be  expressed  by  AB  ; 
MP  =  the  horizontal  distance  =  J/  cos  n  =  D  ; 
AB  =  a. 


_J  -rj 


FIG.  4. 
Now  the  angle 

BA  G  =  90  +  (n  —  m)  • 
...  ABG  =  90  -  (n  +  m)  ; 
AF  sin  7?i 


Hence 


or, 

and 

or. 


GF      sin  [90  +  (n  —  m)] ' 
AF  =    GFsmm 

cos(w  —  m)' 
BF  sin  7/1 


GF      sin  [90  -  (71  +  m)] 
GF  sin  771 


cos  (n  +  m) 

.-,  AF+  BF=GFs'mm[ 
But     AF+BF=a, 
and  C 


Lcos(7i  —  m) 
CD  cos  7/1 


cos  (71  +  m)_ 


2  tan  tyi        2  sin  m 


THE    STADIA.  V 

Substituting  this  value  of    GF  in  the  equation  above,  we 
obtain 

CD  cos  m  [cos  (n  +  m)  +  cos  (n  —  m)] . 


*  -^ 


2  [cos  (n  +  m)  cos  (n  —  m)] 
cos2  n  cos2  m  —  sin2  n  sin2  m 

cos  n  cos2  ??7 
cos2n  cos2m  —  sin2n  sin2m 


and  IX  =  c  +  ka 

cosn  cos2m 

Whence  D  =  c  cosn  +  fca  cos2w  —  ka  sin'2n  tan2m. 

The  third  term  of  the  second  member  of  this  equation  may 
be  safely  neglected,  as  it  is  very  small,  even  for  long  distances 
and  large  angles  of  elevation  (for  1500',  n  =  45°  and  k  =  100, 
it  is  but  0.07')  ;  therefore,  the  final  formula  for  distances,  with 
a  stadia  kept  vertical,  and  with  wires  equidistant  from  the 
center  wire,  is  the  following  :  — 

D  =  c  cos  n  +  ak  cos  %.  (2) 

The  value  of  c  cos  n  is  usually  neglected,  as  it  amounts  to 
but  1  or  1.5  feet ;  it  is  exact  enough  to  add  always  1.25'  to  the 
distance  as  derived  from  the  formula, 

D  =  ak  cos  2n.  (2a)* 

14.  The  focal  length  of  the  object-glass  may  be  found  by 
focussing  the  instrument  upon  some  distant  object,  say  a  heav- 
enly body,  and  measuring  then  the  distance  between  the 
plane  of  the  cross-wires  and  that  of  the  objective.  ON,  being 
equal  to  the  distance  between  the  objective  and  the  intersec- 
tion of  a  plumb-line  with  the  horizontal  axis  of  the  telescope, 
may  be  obtained  by  direct  measurement.  The  distance  p, 
between  the  stadia  wires,  may  be  determined  as  follows  :  — 

Set  up  the  instrument  on  level  ground,  and  measure  forward 
from  the  pLumb-line  a  distance  equal  to  c,  and  mark  the  point ; 
measure  onward  from  the  mark  any  convenient  distance  d,  400 
or  500  feet  as  a  base.  The  telescope  being  level,  observe  care- 
fully the  space  a  intercepted  by  the  stadia  wires  on  a  leveling 

*  The  above  explanation  of  the  stadia  is  substantially  that  given  by 
Mr.  G.  J.  Specht,  published  by  Van  Nostrand,  1884,  though  corrected 
and  simplified.  See  Table  XIX  for  reducing  stadia  measurements. 


10  PRELIMINARY    SURVEY. 

rod  held  vertically  at  the  farther  extremity  of  the  base.  Then 
from  the  proportion  d  — f :  a  ==f :  p,  the  required  distance  p 
may  be  obtained. 

EXAMPLES. 

1.  Given  f=  8  inches,  base  =  500  feet,  and  a  =  5.25  feet. 
Find  p  =  .084  inches. 

2.  At  what  fractional  part  of  the  focal  length  must  the 
stadia  wires  be   separated   so  that  one  foot  on  the  rod  will 
correspond  to  100  feet  base  V 

2°.    The  Gradienter. 

15.  The  gradienter  is  an  attachment  to  the  transit  which 
may  be  used  for  running  grades,  determining  distances,  etc. 
In  its  construction,  a  clamping  arm  extends  downwards  from 
the  axle  upon  which  the  telescope  revolves,  and  is  forked  at 
lower  extremity  to  embrace  a  micrometer  headed  nut.  This 
nut  moves  along  a  screw  accurately  cut,  making  a  certain 
number  of  revolutions  to  the  hundredth  of  a  foot.  The  head 
of  the  screw  is  graduated  into  one  hundred  parts,  and  attached 
is  a  zero  edge  for  reading  the  graduations. 

As  the  proportion  of  the  screw  is  such  that  a  complete 
revolution  gives  one  foot  vertical  in  a  distance  of  100  feet 
horizontal,  when  the  motion  of  the  telescope  measures  this 
foot,  it  necessarily  follows  that  the  rod  must  be  100  feet 
distant ;  or  if  telescope  measures  1.5  feet  the  rod  must  be  150 
feet  distant. 

Hence,  to  run  a  certain  gradient,  bring  the  telescope  level  by 
means  of  the  milled  head  screw,  and  note  the  reading  ;  then  continue 
the  motion  of  the  milled  head  one  revolution  and  part  thereof  for 
each  foot  and  part  thereof,  of  foot  per  hundred  of  the  desired 
gradient. 

Thus,  to  set  off  a  gradient  of  0.5  foot  per  100  feet,  move 
micrometer  milled  head  50  graduations  from  the  level. 

To  set  off  1.25  foot  per  100  feet,  move  one  revolution  and 
twenty-five  graduations. 


THE    VERNIER. 


11 


To  measure  distances,  note  the  space  on  rod  passed  over  by 
one  revolution  of  the  micrometer  head.  Thus,  if  one  revolution 
of  micrometer  head  passes  from  4.2  to  5.6,  the  difference,  1.4, 
indicates  the  distance  of  rod  from  instrument  of  140  feet. 


3°.    The  Vernier. 

16.  The  Vernier.  Though,  perhaps,  it  cannot  be  con- 
sidered an  auxiliary  to  the  transit  in  the  same  sense  as  the 
preceding,  it  is  thought  best  to  give  in  this  place  some 
explanation  of  it,  more  especially  for  the  benefit  of  the 
young  engineer. 

It  is  an  auxiliary  scale  for  measuring  smaller  divisions  than 
those  into  which  a  graduated  scale  or  limb  is  divided.  The 
smallest  reading  of  the  vernier,  or  least  count,  is  the  difference 
in  length  between  one  division  on  the  graduated  scale  or  limb, 
and  one  on  the  vernier.  If  the  divisions  on  the  vernier  are 
smaller  than  those  on  the  limb,  the  vernier  is  direct;  if  the 
reverse,  retrograde. 


10 


11 


12         13 


15 


16 


17 


V 

R 

0123           456788         10 

FIG.  5. 

Let  LM  represent  any  scale  divided  into  tenths,  and  we  wish 
to  measure  or  read  to  tenths  of  these  divisions,  i.  e.,  to  T^. 

Using  a  direct  vernier,  we  should  have  10  spaces  on  it  equal 
to  9  on  the  scale,  and  each  one  of  them  equal  to  -^  of  T^,  or 
y^,  of  the  scale  of  graduation,  giving  a  least  count  of  -^^ 
-  Tim  =  Ttj<y>  as  desired. 

To  read  to  twentieths  of  the  divisions  on  the  scale,  w^e 
should  have  20  divisions  on  the  vernier  corresponding  to  19 
on  the  scale,  or  ^each  space  on  the  vernier  equal  to  £$  X  -^ 
=  *W  and  Siving  a  least  count  of  inT0<y  —  =  - 


12  PRELIMINARY    SURVEY. 

In  general,  if  s  =  the  smallest  division  of  the  scale  or  limb, 
v  =  the  smallest  division  of  the  vernier, 
n  —  the   number  of  divisions  on   the  vernier, 

we  shall  have  least  count  =  s  —  v  =  ^. 

Or,  the  least  count  of  a  vernier  is  equal  to  the  smallest 
division  of  the  scale  or  limb  divided  by  the  number  on  the 
vernier.  * 

If  s  =  \  degree,  and  n  =  30,  as  ordinarily  found  on  transit 
plates,  the  least  count  will  be  ^  4-  30  =  ^  of  a  degree  =  one 
minute. 

If  s  =  ^  degree,  and  n  =  40,  as  sometimes  found  on  vertical 
arcs  to  solar  attachments,  the  smallest  reading  =  1  -^-  ?1^=T^Tj 
of  a  degree  =  \  minute. 

To  space  a  vernier  for  a  given  least  count,  say  10"  on  a 

limb  graduated  to  10',  we  must  have  n  =  —  =  10  -j-  i  =  60 

s—v 

spaces,  covering  59  spaces  on  the  limb. 

17.  To  read  an  instrument  having  a  vernier  consists  in 
determining  the  number  of  units  and  fractional  parts  thereof, 
into  which  its  scale  or  limb  may  be  divided,  from  the  zero 
point  on  the  limb,  where  the  graduation  begins,  to  the  zero 
point  of  the  vernier. 

It  is  accomplished  as  follows:  Take  the  reading  of  the  scale 
or  limb,  as  shown  by  the  last  graduation  preceding  the  zero  of 
the  vernier;  then  find  a  line  on  the  vernier  which  coincides 
with  a  line  on  the  scale  or  limb.  The  number  of  this  line,  as 
indicated  by  the  graduation  on  the  vernier,  shows  how  many 
units  of  the  least  count  are  to  be  added  to  the  first  reading. 

EXERCISES. 

1.  An  arc  is  graduated  into  quarter-degrees,  and  a  vernier 
of  30  parts  covers  29  parts  of  the  arc;  find  the  least  count. 

2.  Design  a  vernier,  which,  when  applied  to  a  limb  gradu- 
ated into  20',  will  give  a  least  count  of  20".  f 

*  It  is  evidently  immaterial  whether  LM  be  straight  or  curved. 
t  The  foregoing  description  of  the  vernier  is  taken  from  the  author's 
Treatise  on  Plane  Surveying. 


THE    TRANSIT    PARTY.  13 

18.  In  running   a  long  tangent,  or  prolonging  a  straight 
line  with  the  transit,  the  instrument  should  be  in  good  adjust- 
ment ;  it  should  be  properly  centered,  that  is,  set  precisely  over 
the  center  of  the  station  from  which  the  observation  is  to  be 
made,  especially  if  the  point  to  be  sighted  —  back  sight  —  is 
near  the  observer.     The  error  arising  from  an  eccentric  setting 
is  inversely  as  the  distance  of  the  object  sighted  ;  an  eccentric 
setting  of  one   inch  producing  an  error  of  nearly  three  (3') 
minutes   of  arc   in  sighting   100  feet,  while  the  error  arising 
from  a  sight  of  900  feet  is  less  than  one-third  (£')  of  a  minute. 
The  instrument  should  be  level,  especially  across  the  line  of 
sight.     The  sight-pole  should  be  held  plumb,  and  exactly  on  the 
proper  point.     The  observation  should  be  made  as  near  the 
bottom   of   the  sight-pole   as  possible  ;    the  line  of   sight  as 
nearly  horizontal,  and  the  range  in  both  directions  as  nearly 
equal,  as  practicable. 

When  a  well-defined  distant  object  can  be  sighted  ahead,  it 
is  better  to  set  the  instrument  by  it  than  to  trust  to  a  back- 
sight. 

When  great  accuracy  is  required,  errors  of  adjustment  may 
be  lessened  by  reversing  the  instrument  in  altitude  and  azimuth, 
making  two  observations  at  each  station,  and  taking  the  mean 
of  their  readings. 

The  transit  notes  are  written  from  the  bottom  of  the  page 
upwards,  analogous  to  the  "  column  form  "  generally  used  in 
surveying  land.  The  left-hand  page  of  the  transit  note-book 
is  usually  prepared  for  this  purpose,  while  the  right-hand  page 
is  suitably  ruled  for  recording  some  more  details  of  the  work 
as  it  may  be  necessary. 

19.  The  chief  of  the  field  corps  directs  the  operations  of 
the    party,    provides    accommodations    and     subsistence,    and 
pays  the  necessary  expenses.     He  indicates  the    direction   of 
the    line,    establishes    the    deflection    points,    selects    suitable 
sites  for    the    crossing    of    streams,  being    careful    always  to 
run  the  line  as    nearly    as   his    judgment   dictates,  and  with 
the  minimum  grade  in  view,  over  ground  likely  to  be  chosen 
for  location. 


14  PRELIMINARY    SURVEY. 

On  preliminary  work,  especially  in  settled  districts,  the 
clearing  should  be  kept  at  a  minimum,  that  growing  crops  and 
forests  be  injured  as  little  as  possible. 

The  chief  verifies  and  supplements  the  work  of  reconnoissance, 
he  observes  the  quality  of  material  to  be  moved,  and  the 
timber  and  rock,  with  a  view  of  using  them  in  the  construction. 
He  should  be  considerate  of  the  rights  of  landholders  and  not 
do  or  allow  anything  to  be  done  by  any  member  of  the  corps 
which  would  tend  to  arouse  active  opposition  to  the  project, 
but,  on  the  contrary,  by  due  regard  endeavor  to  secure  their 
aid. 

20.  The   transitman  keeps  his  instrument    in    adjustment, 
observes  the  direction  of  the  line  either  by  needle  or  plates, 
keeps  the  axemen  in  line  when  clearing,  and  the  chainmen 
when  measuring  ;  he  notes  the  directions  and  names  of  the 
principal  highways  and  streams  intersected,  and,  when  practi- 
cable, property  lines  with  the  names  of  the  owners.     He  records 
also  the  lengths  of  the  lines  run. 

21.  The  head  chainman  when  measuring  advances  with  the 
chain  and  a  sight  pole,  and  being  put  in  line,  usually  at  a 
chain's    length,  by  the  transitman,    directs    the  stakeman  to 
drive  a  stake  there.     In  a  wooded  district  or  brush  land  where 
much  clearing  is  to  be  done,  the  head  chainman  should  aid 
the  transitman  in  giving  line  to  the  axemen,  by  going  ahead 
and  ranging  it  out. 

22.  The  rear  chainman  should  see  that  his  end  of  the  chain 
is  held  at  the  proper  point,  and  that  the  chain  is  horizontal  and 
taut  when  the  head  chainman  is  setting  the  next  succeeding 
stake.     If,  for  any  reason,  a  portion  of  the  line  is  run  without 
stakes,  and  pins  are  used,  he  should  keep  the  tally.     He  should 
see  that  the  numbers  placed  upon  the  stakes  by  the  stakeman 
indicating  the   stations  are  correct,  and  with  this  in  view  he 
should  call  out  the  numbers  of  the  stations  as  he  approaches 
them. 

Should  a  change  be  made  in  the  direction  of  the  line,  he 
should  note  mentally  the  plus  if  any,  and  be  careful  that  the 


THE    LEVEL    PARTY.  15 

next  stake  is  set  in  its  proper  place,  that  the  uniform  distance 
between  stations  may  be  preserved.  He  should  be  provided 
with  a  book  in  which  to  record  turning  points,  the  intersection 
of  streams,  highways,  and,  when  practicable,  property  lines. 

23.  The  axemen  do  the  necessary  chopping  and  clearing  the 
way,  so  that  the  transitman  and  leveler  may  have,  if  practicable, 
unobstructed  sight,  consistent  however  with  the  directions  to 
the    chief    of   party,   and    for    economic   reasons    keeping   the 
width  of  the  cutting  a  minimum. 

24.  The  stakeman  marks  or  numbers  the  stakes  and  drives 
them  vertically  at  points  indicated  by  the  head  chainman,  with 
the  numbers  so  that  they  can  be  read. by  the  rodman  and 
topographer  as  they  advance  along  the  line.     If  stakes  are  not 
provided  the  party,  it  is  the  duty  of  the  stakeman  to  keep 
himself  supplied,  using  the  proper  means  at  hand.    If,  however, 
but  little  or  no  clearing  is  to  be  done,  an  axeman  should  be 
detailed  to  keep  up  the  supply. 

If  the  deflections  of  the  lines  are  determined  by  the  plates,  a 
back  flagman  is  required  to  give  position  of  last  transit  point, 
but  his  services  will  not  be  needed  if,  as  is  sometimes  the  case, 
the  bearings  are  taken  directly  by  needle. 

B.    THE    LEVEL    PARTY. 

25.  The  level  party  consists  of  a  leveler  and  a  rodman,  and 
the  instruments  needed  are  a  level,  a  rod,  and  a  small  axe. 

THE  Y  LEVEL. 
The  principal  adjustments  are 

1.  The  Line  of  Collimation. 

2.  The  Level  Bubble. 

3.  The  Wyes. 

26.  To  adjust  the  line  of  collimation,  or  to  make-  it  coincide 
with  the  optical  axis  of  the  telescope. 

Set  up  the  instrument  firmly,  remove  the  pins  from  the 
clips,  clamp  to  spindle,  and  bring  the  intersection  of  the  cross- 


16  PRELIMINARY    SURVEY. 

hairs  upon  a  well-defined  point  a  few  hundred  feet  distant. 
Then  carefully  turn  the  telescope  half-way  round  in  the  wyes  ; 
the  bubble-tube  will  then  be  above  the  telescope.  Observe 
again  the  point,  and  see  if  the  intersection  of  the  cross-hairs  is 
still  on  it.  If  it  be  not,  then  bring  either  or  both  cross-hairs,  as 
may  be  required,  half-way  back,  using  the  capstan-head  screws 
perpendicular  to  the  one  which  it  is  desired  to  move. 

27.  To  adjust  the  level  bubble,  or  to  make  the  axis  of  the 
bubble-tube  parallel  to  the  longitudinal  axis  of  the  telescope. 

a.  Clamp  the  telescope  over  either  .pair  of  leveling  screws, 
and  bring  the  bubble  to  the  center  of  its  run.     Turn  the  tele- 
scope in  the  wye's,  so  as  to  bring  the  bubble-tube  a  little  to 
either  side  of  the  center  of  the  bar.     If  the  bubble  runs  towards 
either  end,  bring  it  back  to  the  center  by  the  capstan-head 
screws,  which  are  set  in  either  side  of  the  tube-holder. 

Again  bring  the  bubble-tube  to  the  center  of  the  bar  and 
the  bubble  to  the  center  of  its  run  ;  turn  the  tube  to  either 
side,  and  repeat  the  correction  if  necessary  until  the  bubble 
will  keep  its  position  when  its  tube  is  turned  half  an  inch  or 
more  on  either  side  of  the  bar. 

The  necessity  for  this  operation  arises  from  the  fact  that, 
when  the  telescope  is  reversed  end  for  end  in  the  wyes,  in  the 
other  and  principal  adjustments  of  the  bubble,  we  are  not 
certain  of  placing  the  bubble-tube  in  the  same  vertical  plane  ; 
and,  therefore,  it  would  be  almost  impossible  to  effect  the 
adjustment  without  a  lateral  correction. 

b.  Now  bring  the  bubble  to  the  centers  of  its  run,  and  with- 
out jarring  the  instrument  take  the  telescope  out  of  the  wyes 
and  reverse  it  end  for  end.     Should  the  bubble  run  to  either 
end,  bring  it  half-way  back  by  lowering  that  end  or  raising 
the  other,  using  the  capstan-head  screws   at  the  end  of  the 
tube.     Verify  the  adjustment. 

28.  To  Adjust  the  Wyes,  or  to  make  the  axis  of  the  bubble- 
tube  perpendicular  to  the  vertical  axis  of  the  instrument. 

Clamp  the  telescope  in  the  wyes  ;  release  from  spindle,  place 
the  telescope  over  one  pair  of  leveling  screws  and  bring  the 
bubble  to  the  center  of  its  run  ;  then  turn  the  telescope  hori- 


LEVELING.  17 

zontally  half-way  round.  If  the  bubble  runs  towards  either 
end  bring  it  half-way  back  by  the  adjusting  screws  at  the  end 
of  the  bar,  and  one-half  by  the  leveling  screws.  Proceed  in 
the  same  manner  with  the  telescope  for  the  other  pair  of  level- 
ing screws.  Repeat  the  operation. 

29.  A  surface  like  that  of  still  water  may  be  called  a  level 
surface.     The  curve  formed  by  the  intersection  with  such  a 
surface  of  a  vertical  plane  is  a  line  of  true  level;  a  line  tangent 
to  the  latter  is  a  line  of  apparent  level. 

Leveling  is  the  art  of  determining  the  differences  of  eleva- 
tion of  two  or  more  points,  or  of  determining  how  much  one 
point  is  above  or  below  a  line  of  true  -level  passing  through  the 
other  point. 

30.  From  the  foregoing  it  is  evident  that,  on  account  of  the 
curvature  of  the  earth,  a  horizontal  line  is  not  really  through- 
out its  length  a  level  line;  that  of  two  points  in  the  same  level 
line  each  will  have  its  own  horizon.     Hence  in  leveling  the 
effect  of  the  curvature  of  the  earth  upon  the  comparative  ele- 
vations of  different  points  must  be  taken  into  consideration. 
The  effect  of  the  curvature  is  to  make  objects  appear  lower 
than  they  really  are.     The  air  nearer  the  surface  of  the  earth 
is  denser  than  that  farther  removed  from  the  surface.     This 
difference  in  density,  causing  refraction  of  light,  will  affect  the 
elevation  of  a  point  as  observed  through  the  telescope  of  a 
level,  so  that  it  also  must  be  taken  into  consideration.     Its 
effect  is  to  make  objects  appear  higher  than  they  really  are. 
The  error  caused  by  refraction  is  one-seventh  as  great  as  that 
caused  by  curvature. 

Let  us  first  find  an  expression  for  the  correction  due  to  the 
curvature  of  the  earth.  That  is  — 

31.  To  find    the    deviation  from    its  tangent  of  a  line  of  true 
level. 

Let  0  represent  the  center  of  the  earth,  PN  a  line  of 
true  level,  and  PN'  its  tangent,  or  a  line  of  apparent  level. 
The  distance  NN'  corresponding  to  the  length  of  sight  PN 
is  required. 


18 


PRELIMINARY    SURVEY. 


O 


From  Geometry, 

PN'2  =  NN'  (2  ON  +  NN'); 


or, 


,  PN 


2  ON  +  NN' 


FIG.  6. 


For  ordinary  distances,  the  length 
of  the  arc  may  be  regarded  as  that  of 
the  tangent,  and  NN7  as  inconsider- 
able in  comparison  with  2  ON,  the 
diameter  of  the  earth.  Therefore,  call- 
ing the  length  of  sight  rf,  the  correc- 
tion c,  and  the  radius  of  the  earth  r, 
we  have 


and  the  correction  for  refraction 


=  - 

2r       14r' 


(4) 


then  the  correction  due  to  curvature  and  refraction,  wrhich  we 
will  call  C,  is 


or, 


7°      2r      Ur' 


Ir 


(4a) 


This  correction  must  be  added  to  the  height  of  the  object  as 
found  by  the  level. 

In  practice,  the  necessity  for  using  the  above  formula  is 
avoided  whenever  it  is  possible  to  set  the  level  at  equal 
distances  from  the  points  whose  difference  of  height  is 
required. 

EXERCISES. 

1.  Assuming  the  diameter  of  the  earth  7,926  miles,  show 
that  for  a  mile  sight  c  =  about  8  inches.  Find  the  value  of  C 
for  the  same  distance. 


LEVELING. 


19 


2.  What  is  the  correction  due  to  curvature  for  half  a  mile  ? 
Two  miles  ? 

3.  What  is  the  length  of  sight  when  C  equals  one-tenth  of  a 
foot? 

4.  Show  that,  practically,   the  correction   for  curvature   in 
feet  is  equal  to  two-thirds  the  square  of  the  distance  in  miles. 


32.  If  two  points  M,  N,  whose  difference  of  elevation  is 
required,  can  be  observed  upon  from  some  point  P  about  equi- 
distant* from  them,  not  necessarily  in  their  line,  set  up  the  level 
at  P,  and  note  the  reading  of  a  rod  held  vertically  over  each 
point.  The  difference  of  the  two  readings  will  indicate  the 
difference  of  level  required. 


FIG.  8. 


33.  If  the  above  method  is  impracticable,  set  up  the  instru- 
ment at  some  point  P  —  either  in  or  out  of  the  line,  no  matter 
which  —  from  which  a  rod  maybe  observed  on  the  first  station 
M,  and  also  on  another  point  0,  in  the  direction  of  JV,  about 
equidistant  with  M  from  the  instrument.  Remove  the  level 
to  a  new  position  P',  whence  observe  again  the  rod  on  O,  also 
the  rod  reading  at  N. 

The  difference  between  the  readings  of  the  rod  at  M  and  0 
shows  how  much  higher  the  latter  is  than  the  former,  and  in 
like  manner  the  difference  of  the  readings  at  0  and  N  gives 


*  Placing  the  instrument  in  this  position  lessens  the  effects  of  inaccu- 
rate adjustment  and  renders  unnecessary  the  corrections  indicated  in 
Article  31. 


20  PRELIMINARY    SURVEY. 

the  difference  in  elevation  of  these  points,  and  so  on,  no  matter 
what  the  number  of  stations.  The  difference  in  height  of  M 
and  TV 

=  M m  —  Oo  +  Oo'  —  Nn  ; 
or,  Mm  +  Oo'  —  Oo  —  Nn 

=  Mm  +  Oo'  -  (Oo  +  Nn). 

Calling  Mm  and  Oo'  back-sights,  and  the  other  two,  fore- 
sights, we  perceive  that  the  difference  of  level  of  two  points  is 
shown  by  subtracting  the  sum  of  the  fore-sights  from  the  sum 
of  the  back-sights. 

34.  Again,  in  leveling,  we  measure,  by  means  of  the  rod, 
how  much  lower  than  the  line  of  sight  (height  of  instrument) 
certain  points  are.     Thus  we  may  determine  the  relative  eleva- 
tions of  the  points.     Suppose,  for  example,  it  be  required  to 
determine  the  difference  in  elevation  of  any  two  points.     For 
reasons  already  given,  set  the  level  equally  distant  from  the 
points.     If  this  cannot  be  done,  and  both  observations  have  to 
be  taken  from  one  of  the  stations,  especially  if  the  distance 
between    them    is    considerable,  correction    as    previously  de- 
scribed must  be  made.    But  in  this  case  suppose  it  is  possible  ; 
and  suppose  that  when  held  on  one  point,  the  rod  reads  7.255  ; 
that  is,  this  point  may  be  considered  7.255  below  the  line  of 
sight,  and  4.755  when  held  on  the  other  ;    then  the  first  may 
be  considered  7.255  —  4.755,  or  2.500  farther  than  the  second 
below  the  line  of  sight,  or  lower  than  the  second. 

35.  Suppose  it  be  required  to  determine  the  difference  in 
elevation  between  two  points,  of  which  one  is  so  much  higher 
than  the  other  that  the  rod  is  too  short  to  give  a  reading  011 
both  points  for  one  position  of  the  instrument.     In  such  a  case 
one  or  more  auxiliary  points,  called  turning  points  (T.  P.),  must 
be  used,  and  their  relative  elevations  determined.     Suppose  the 
reading  on  the  first  point  is  0.824,  and  on  a  turning  point  is 
10.432  ;  the  latter  is  then  9.608  below  the  former.     Now  the 
instrument  must  be  moved  and  set  up  so  as  to  obtain  a  reading 
on  the  turning-point,  and  (we  will  suppose)  on  the  other  of 
the  given  points. 


LEVELING.  21 

Suppose  that  on  the  former  it  is  1.302,  and  on  the  latter 
8.634  ;  the  latter  is  then  7.332  below  the  turning-point,  or 
9.608  +  7.332,  or  16.940  below  the  first  of  the  two  given 
points. 

The  first  sight  taken  after  setting  up  the  level  is  called  a 
back-sight,  or  plus  sight  ;  those  taken  after  this,  and  before 
the  instrument  is  moved,  are  called  fore-sights,  or  minus  sights. 
As  the  difference  of  the  readings  of  the  rod  on  two  points 
gives  their  difference  of  elevation,  so  the  difference  of  the  sum 
of  the  plus  sights,  and  the  sum  of  the  minus  sights  on  T.  P.'s 
and  the  last  point  will  give  the  difference  in  elevation  of  the 
extreme  points.  In  the  above  example. 

0.824  10.432 

1.302  8.634 

2.126  19.066 

19.066  -  2.126  =  16.940,  as  before. 

This  is  used  as  a  check  on  level-notes. 

In  extended  leveling,  permanent  elevations,  fixed  during  the 
progress  of  the  work  for  future  reference,  are  called  bench 
marks,  or  benches  (B.  M.). 

36.  In  leveling,  it  is  customary  to  refer  all  elevations  to  an 
assumed  level  plane,  called  the  plane  of  reference,  the  datum 
plane,  or  simply  the  datum.  Points  are  then  said  to  be  so 
much  above  or  below  the  datum.  As  this  plane  may  be 
assumed  at  pleasure,  it  is  generally  so  taken  as  to  be  lower 
than  any  point  whose  elevation  is  to  be  determined. 

If  the  beginning  of  a  survey  is  in  the  vicinity  of  tide-water, 
this  plane  is  assumed  at  the  height  of  mean  low  water,  which 
elevation  may  be  called  zero.  Then  a  point  which  has  the 
elevation  125.37  will  be  125.37  above  low  water. 

If  two  points  have  the  elevations  125.375  and  105.213,  re- 
spectively, the  former  is  125.375  — 105.213,  or  20.162  higher 
than  the  latter. 

The  datum  having  once  been  determined,  its  elevation,  or 
that  of  a  point  a  known  distance  above  it,  should  be  perma-   j 
nently  fixed  for  future  reference  and  comparison. 


PRELIMINARY    SURVEY. 


37.    There  are  various  forms  employed  for  recording  level 
notes.     The  following  is  simple  and  convenient  :  — 


T.P 


T.P. 


STA. 

+  s. 

H.I. 

-S. 

ELEV. 

KKMARKS. 

B.M. 

4.725 

100.000 

On  S.  end,  lower 

step,  University, 

0 

104.725 

2.44 

102.285 

Main  Bldg. 

1 

1.25 

103.475 

+60 

8.417 

.50 

104.225 

Pt.  on   sidewalk, 

S.  of  Main  Bldg. 

2 

112.642 

7.80 

104.842 

3 

6.50 

106.142 

4 

4.28 

108.362 

5 

3.365 

1.36 

111.282 

0 

114.647 

1.25 

113.397 

7 

5.45 

109.197 

The  bench  mark  is  assumed  to  be  100  feet  above  the  datum. 
The  first  plus  sight  is  4.725,  which  added  to  100  gives  104.725, 
the  height  of  the  instrument  (H.  T.)  above  the  datum.  The 
first  minus  sight  is  taken  on  station  0,  and  is  2.44,  which  sub- 
tracted from  104.725  gives  102.285,  the  height  of  this  station 
above  the  datum.  Similarly,  the  height  of  station  1  and  plus 
60  are  obtained,  the  latter  being  a  turning  point.  The  instru- 
ment is  then  carried  forward,  set  up  again  in  a  convenient 
place  to  sight  the  T.P.  and  other  points  in  the  line,  and  thus 
the  work  proceeds. 

At  one  setting  of  the  instrument  the  elevations  of  points, 
besides  the  turning  points,  which  are  not  too  high  or  too  low 
to  be  reached,  may  be  ascertained.  It  is  evident  that  if  any 
error  be  made  at  a  T.  P.,  all  the  following  elevations  will 
thereby  be  affected  ;  but  if  made  at  one  of  the  other  points, 
only  the  elevation  of  that  point  will  be  affected.  Hence  the 
importance  of  careful  observations  at  the  T.  P.'s. 

38.  Wind  and  sunshine  affect  the  accuracy  of  leveling  as 
of  work  with  the  transit.  For  very  good  work  it  is  desirable 
to  have  a  calm  day  on  which  the  sun  is  obscured  by  clouds. 
In  addition  to  a  proper  manipulation  of  the  instrument,  the 
sights  should  not  exceed  300  feet,  the  rod  should  be  held  ver- 


THE   TOPOGRAPHIC   PAKTY.  23 

tical,  and  the  rodman  should  select  for  turning-points  good  and 
firm  points  on  stones,  pegs,  etc.,  on  which  the  rod  may  be 
freely  turned  or  spun  around.  Test  the  arithmetical  work  in 
the  foregoing  table  as  indicated  in  Art.  35. 

39.  The  leveler  keeps  his  instrument  in  adjustment,  sights 
the  rod  when  held  on  stations,  turning  points,  or  benches  of 
the  rodman,  and  records  his  readings  in  a  book  provided  for 
the  purpose.     He  ordinarily  takes  a  bench  reading  every  1500 
or  2000  feet,  and  oftener  in   a  hilly  district.     If  the  benches 
are  judiciously  chosen  they  may  frequently  serve  as  turning- 
points,  arid  a  saving  of  time  be  thereby  effected. 

The  rod  should  be  read  to  tenths  of  a  foot  on  intermediates 
and  to  thousandths  of  a  foot  on  benches  and  turning  points. 

The  leveler  should  observe  the  surface,  and  when  practicable 
the  high  water  mark,  of  creeks  and  rivers.  He  should  use  a 
Locke  level  to  take  the  heights  when  crossing  deep  gulches  or 
narrow  ravines,  and  thus  save  the  time  required  to  peg  in  the 
usual  way.  The  profile  should  be  made  up  daily. 

40.  The  rodman  holds  his  rod  vertically  for  observation  by 
the  leveler  at  every  stake,  the  number  of  which  he  calls  out  to 
him,  and  wherever  there  is  an  observation  to  be  taken  on  a 
plus,  for  instance  on  bank  or  in  bed  of  stream,  he  should  make 
known   to   the  leveler  its  amount.     He   should   be  quick  to 
perceive  a  singular  point,  and  prompt  to  decide  whether  a  plus 
observation  is  required  on  it.     He  should  note  the  position  of 
benches  and  turning  points,  so  as  to  find  them  readily,  should 
it  be  desired  to  re-level  the  line.     He  should  assist  the  leveler 
when  required  to  make  up  the  profile. 

C.    THE   TOPOGRAPHIC   PARTY.   , 

41.  It  is  the  duty  of  the  Topographic  party  to  indicate  the 
position  of  the  prominent  features  of  a  belt  of  country  extend- 
ing both  ways  from  the  center  line  of  the  survey,  including 
woodland,   streams,  roads,  buildings,  etc.,   and  to  obtain  the 
necessary  data  to  make  a  contour  map  of  it. 

The  party  usually  consists  of  a  topographer  and  two 
assistants.  They  need  a  tape,  rod,  and  an  instrument  for 


24  PRELIMINARY    SURVEY. 

measuring  slopes,  which  may  be  a  clinometer,  cross-section 
pole  (slope  board),  or  a  Locke  level.  A  small  prismatic 
compass  is  sometimes  carried  to  observe  the  direction  of 
objects.  Buildings,  roads,  streams  and  other  objects  are 
usually  located  by  offsets  from  the  center  line. 

42.  The  data  for  the  contour  map  is  obtained  by  observing 
from  every  station  the  slope  of  the  ground  at  right  angles  to 
the  proposed  line,  and  on  both  sides  of  it.*  The  distance 
between  points  of  successive  contours  being  taken  by  the 
assistants,  and  the  total  distance  out  from  the  center  varying 
from  about  50  feet  to  several  hundred  feet,  as  may  be  needed 
by  the  chief  engineer  in  determining  the  final  location. 

It  will  be  perceived  that  the  topographic  work  is  connected 
with  that  of  the  level-party,  and  since  the  elevations  of  the 
stations  are  given  by  the  latter,  the  data  for  a  contour  map 
can  be  easily  obtained. 

For  instance,  if  a  station  in  the  center  line,  as  shown  by  the 
level  notes,  has  an  elevation  of  852.4  feet,  and  five-foot  con- 
tours are  being  taken,  it  is  evident  that  the  850-foot  contour 
passes  2'.4  below  this  point.  Suppose  the  Locke  level  is  used  ; 
then  if  the  observer,  whose  eye  we  shall  assume  is  five  feet 
high,  stands  at  the  station,  and  his  assistant  holds  a  rod  at  a 
point  where  the  line  of  sight  intersects  it  at  7'.4,  the  foot  of  the 
rod  will  be  on  the  850-foot  contour.  Measure  from  the  station 
to  this  point,  and  from  the  point  observe  a  reading  of  ten  feet 
on  the  rod,  which  will  show  the  845-foot  contour.  Measure  to 
this  point,  and  continue  the  observations  until  the  limit  of 
measurement  in  this  direction  is  attained.  Proceed  in  a 
similar  manner  to  take  the  slope  of  the  upper  side,  and  so  pass 
along  the  entire  line.  Otherwise  the  observer  places  himself 
where  the  rod  reading  on  the  station  is  2.6  feet,  then  his  line 
of  sight  will  indicate  the  855-foot  contour;  he  evidently  stands 
on  the  850-foot  contour,  and  without  changing  place  he  dis- 
covers the  845-foot  contour  where  the  rod  reads  10  feet. 

*  Sometimes  on  preliminary  work,  and  especially  where  the  ground 
is  gently  undulating,  the  contours  are  taken  from  stations,  200  or  300 
feet  apart. 


THE    TOPOGRAPHIC    PARTY.  25 

Measure  from  the  station  and  locate  the  points  as  above.  In 
a  similar  manner  the  Topographer  will  stand  on  the  840-foot 
contour  when  sighting  the*  bottom  of  rod  held  on  the  845- 
foot  contour,  whence  a  rod  placed  so  as  to  give  a  reading  of 
10  feet  will  indicate  the  835-foot  contour.  In  a  correspond- 
ing manner  the  contours  on  the  upper  side  of  the  line  may  be 
shown.  If  the  height  of  the  observer's  eye  is  not  5  feet  pro- 
ceed in  general  as  though  using  the  wye  level.  The  record 
may  be  kept  in  fractional  form,  the  numerator  indicating  the 
contour,  and  the  denominator  the  distance  of  the  observed 
point  in  it  from  the  center  line. 

In  a  similar  manner  a  simple  cross-section  at  any  station  can 
be  obtained  j"  the  height  of  the  eye  giving  the  difference  in 
elevations  at  the  successive  points  of  observation,  the  distances 
between  these  being  measured  by  the  tape.  If  a  cross-section 
pole  (called  also  a  slope  board)  and  leveling  rod  are  used,  the 
pole  is  held  level  and  the  rod  indicates  the  difference  in  height 
of  its  ends,  while  the  length  of  the  pole,  usually  10  or  12  feet, 
gives  the  distance  between  the  successive  readings.  Cross- 
sections  may  be  taken  very  rapidly  by  either  of  these  methods. 

In  the  office  the  contour  map  is  made  by  connecting  the 
points  of  equal  height,  and  writing  the  elevation  on  the 
several  lines. 

Let  Fig.  9  represent  a  portion  of  a  contour  map  drawn  to  a 
scale  of  400  feet  to  the  inch.  We  will  suppose  the  line  nearest 
the  top  of  page  to  be  100  feet  above  datum  or  the  100-foot 
contour,  and  the  farthest  one  the  130-foot  contour.  The  differ- 
ence between  the  successive  contours  being  5  feet.  The  dotted 
line  represents  the  contour  of  the  grade  rising  one  foot  per 
station,  or  a  one  per  cent,  grade,  from  L  to  N.  If  this  line  be 
adopted  for  the  location,  there  would  be  neither  center-cut  nor 
center-fill.  If  the  straight  line  LN  is  adopted,  the  plan  shows 
that  there  would  be  a  center-cut  at  b  of  about  11  feet,  one  at 
g  of  3  feet  ;  and  a  center-fill  at  e  of  14  feet,  but  at  d  and/ 
there  would  be  neither  cut  nor  fill,  hence  these  are  grade  points. 
The  cuts  and  fills  at  the  center  of  the  line  being  shown  by  the 
number  of  spaces  and  fractions  thereof  between  the  adopted 
line  and  the  grade-contour,  and  hence  a  tolerably  close  approxi- 


26 


PRELIMINARY    SURVEY. 


mation  may  be  made  as  to  quantities.  Table  XIV  will  be 
found  useful  in  this  connection.  While,  therefore,  the  line 
LN  has  the  advantage  of  being  shorter  than  the  dotted  line,  it 
has  the  disadvantage  of  a  heavier  grade,  besides  the  cuts  and 
fills  named. 

Various  conditions  and  circumstances,  it  will  be  readily 
perceived,  present  themselves  to  the  engineer  in  deciding  the 
precise  location  of  a  center  line  of  any  great  extent. 


FIG.  9. 

43.  The  small  plan  above  simply  serves  as  an  illustration, 
but  to  more  fully  comprehend  the  difficulties  surrounding  this 
important  matter  of  location,  one  should  imagine  that  the 
engineer  has  before  him  a  contour  map  of  a  belt  of  country 
several  miles  in  extent,  that  he  is  endeavoring  to  decide  upon 
the  best  location  of  a  line  in  that  belt ;  and  consider  what  are 
the  questions  which  present  themselves  to  him  for  solution, 
and  what  are  his  limitations.  He  desires  to  make  the  line  as 
near  straight  as  practicable  between  certain  points  ;  the  grade 
the  best  possible,  in  no  case  to  exceed  a  certain  amount  ;  the 
cost  of  construction  a  minimum.  It  is  evident  also  that  the 
expense  of  operating  and  maintaining  the  road  is  involved  in 
this  decision.  The  cost  of  the  earth  work  will  in  general  be 


LOCATING   THE    LINE.  27 

lessened  by  equalizing  as  far  as  practicable  the  material  in 
cuts  and  fills.  This,  however,  is  not  always  possible,  because 
of  the  necessity  of  crossing  other  public  highways  or  streams 
at  fixed  grade.  To  lessen  solid  rock  cutting  he  may  consider 
the  question  of  change  of  grade  or  direction  from  what  other- 
wise he  would  deem  the  proper  location.  This  change,  if 
made,  may  necessitate  borrowing  material  elsewhere.  To  secure 
proper  drainage  in  a  flat  and  wet  section,  the  question  of 
waste  of  material  or  its  transportation  a  distance  to  fill  must 
be  settled. 

These,  and  numerous  other  questions  of  a  similar  character, 
present  themselves  in  the  decision  of  this  matter  ;  and  to 
answer  them  properly,  and  thus  make  the  best  location 
possible  under  all  the  conditions  and  circumstances,  an  op- 
portunity is  offered  the  engineer  for  the  display  of  his  best 
judgment. 

In  passing  upon  the  question  of  equalizing  cuts  and  fills,  it 
must  be  remembered  that  rock  measured  in  excavation  will, 
when  broken  and  thrown  in  embankment,  show  an  increase  in 
volume  of  about  two-thirds  ;  that  is  to  say,  3  cubic  yards  of 
solid  rock  in  cut  will  make  about  5  cubic  yards  in  fill.  The 
shrinkage  of  earth  soils,  which  is  about  one-tenth,  is  generally 
ignored  in  this  connection,  though  it  too  should  be  considered 
when  setting  out  the  work  for  the  contractor.  See  Art.  113. 

The  student  may  sketch  a  contour  map  showing  a  descend- 
ing grade  of  H  feet  per  station,  and  determine  the  depth  of 
cuts  and  fills. 

44.  The  engineer  is  thus  enabled  to  make  a  paper  location, 
establish  grades,  and  estimate  approximately  the  quantity  of 
material  to  be  removed.  The  center  line  must  then  be  staked 
out,  curves  run  in  to  connect  the  tangents,  stakes  being  set 
carefully  and  firmly  every  100  feet,  at  the  beginning  and  end- 
ing of  every  curve,  on  the  banks  of  creeks  and  edges  of  ravines 
intersecting  the  line,  and  at  all  other  points  where,  in  the 
judgment  of  the  engineer,  the  work  of  construction  will  be 
expedited  thereby  ;  the  more  important  points  in  the  line 
being  also  referred  to  other  points  at  known  distances  and 


28  PRELIMINARY    SURVEY. 

directions.  These  reference  points  (R.P.'s)  should  be  located 
sufficiently  far  from  the  field  of  operations  that  they  will  not 
likely  be  disturbed  during  the  progress  of  the  work.  The 
levels  must  be  carefully  taken  on  all  points  in  the  line  set  by 
transit,  and  plusses  taken  and  the  rod  read  on  other  points 
in  the  line  where  there  is  a  noticeable  change  in  its  direction 
vertically,  that  a  correct  profile  may  be  made,  and  the  work  of 
cross-sectioning  be  facilitated. 

The  leveler  should  select  his  benches  so  far  from  the  line 
that  they  will  be  undisturbed  during  the  construction,  and 
when  practicable  make  them  nearly  at  grade  for  convenient 
future  reference..  The  rod  should  be  read  to  thousandths  on 
benches  and  turning  points,  but  only  to  tenths  on  intermediates. 

The  width  of  strip,  or  right  of  way,  required  for  railroad 
purposes  is  a  variable  quantity  depending  upon  the  width  of 
the  roadbed  and  the  depth  of  cuts  and  fills.  A  general  rule 
adopted  by  some  roads  is  to  have  30  feet  besides  the  slopes  for 
single  track  and  60  feet  and  slopes  for  double  tracks.  That  is 
to  say,  where  there  is  a  16-foot  cut  with  slopes  1£  :  1  the 
required  width  would  be 

30  +  2  (1|  X  16)  ==  78  feet. 

At  grade  30  feet  would  be  the  width  required  for  single 
track,  and  60  feet  for  double  track.  It  will  readily  be  per- 
ceived that  the  right  of  way  cannot  always  be  figured  from  the 
center  cut  or  fill,  since  there  may  be  a  cut  or  fill  at  the  side 
where  the  center  is  at  grade. 


CHAPTER   III. 


p.c.c. 


SIMPLE    CURVES. 

A.    DEFINITIONS  AND  FUNDAMENTAL  FORMULAS. 

45.    The  center  line  of  a  railroad  is  composed  of  straight 
lines  and  curves.     The  straight  lines  are  called  tangents ;  the 
curves  are  usually  arcs  of  cir- 
cles,   and    are    simple,    com- 
pound or  reversed.  * 

a.  A  simple   curve  is  the 
arc  of  a  circle. 

b.  A   compound  curve    is 
composed  of  two  simple  curves, 
or  branches,  of  different  radii, 
both  lying  on  the  same  side 
of  a  common  tangent  drawn 

at  their  point  of  union,  as  BE,  FlG>  10- 

Fig.  10. 

c.  A  reversed  curve  is  composed  of  two  simple  curves,  or 
branches,  of  the  same  or  different  radii,  lying  on  opposite  sides 
of  a  common  tangent  drawn 

at    their  point    of  union,   as 
BE,  Fig.  11. 

(L  The  point  of  curve,  or 
the  P.O.,  is  the  point  at 
which  the  tangent  AB  ends, 
and  the  curve  BNE  begins, 
as  B,  Fig.  12. 

e.    The  point  of  tangent,  or 
the  P.T.,  is  the  point  at  which  the  curve  BNE  ends,  and  the 
tangent  EF  begins,  as  E,  Fig.  12. 

*  As  a  rule  reversed  curves  are  not  admissible  on  main  line,  but  they 
are  properly  used  in  connection  with  cross-overs,  sidings,  and  in  yai-ds. 


FIG.  11. 


30 


SIMPLE    CURVES. 


/.  The  point  of  intersection,  or  the  P.T.,  is  the  point  of 
intersection  of  the  tangents  drawn  through  the  P.C.  and  the 
P.T.,  as  I. 

g.    The  radius  BO,  or  EO,  is  denoted  by  R. 

h.  The  point  of  compound  curve,  or  the  P.C.C.,  is  the  point 
of  common  tangent  of  its  two  branches.  See  Fig.  10. 

i.  The  point  of  reversed  curve,  or  the  P.R.C.,  is  the  point 
of  common  tangent  of  its  two  branches.  See  Fig.  11. 

j.  The  angle  of  intersection,  or  a-,  indicates  the  amount  of 
divergence  of  the  tangents  BI  and  IE.  Fig.  12. 

k.  The  tangent  distance,  or  T,  is  the  distance  BI  or  El 
from  the  P.C.  or  P.T.  to  the  point  of  intersection. 


46.  Chords  of  one  hundred  feet  are  generally  used,  not  the 
actual  arc,  in  running  a  curve,  and  the  amount  of  curvature  is 
designated  by  the  degree,  though  sometimes  by  the  radius  of 
the  curve. 

47.  The  degree  of  the  curve  D,   as  usually  defined  is  the 
angle  A  OB,  Fig.  13,  at  the  center,  subtended  by  a  chord  as 
AB  of  100  feet  length.     If  it  were  practicable  to  measure  100 
feet   along   the    arc  AB  instead  of  the  chord,  and  substitute 
hence,  in  the  definition  for  degree  of  curve,  the  word  arc  for 
chord,  a  precise  and  convenient  ratio  would  be  always  avail- 
able between  the  radius  and  degree  of  curve;  for  we  should 
then  have 


FUNDAMENTAL    FORMULAS. 


31 


For  a  1°  curve, 


=  100  X  360; 


=  100  X  180;     .-.  7?2  =  m^  =  2864.79. 
2it 

=  100  X  120;     .-.  R3  =  1^P-P_9  =  1909.86. 

2lt 


u       «°         u 


360 


100       .360 


2*JR.  =  100x™;    ,.RH  =  ^.x'™. 


n 


The  inverse  ratio  existing  between  the  degree  of  curve  and 
the  radius  being  apparent.  But  since  the  arc  measurement  is 
impracticable  the  chord  being  substituted  therefor,  and  since  the 
difference  in  length  between  an  arc  and  its  chord  increases  with 
the  degree  of  curve  the  radii  obtained  as  functions  of  the  chords, 
will  not  agree  exactly 
with  those  computed 
as  above.  The  ques- 
tion therefore  arises  to 
whal  extent  is  it  con- 
sistent with  good  prac- 
tice to  assume  the 
equality  of  arc  and 
chord.  In  this  connec- 
tion consider  the  fol- 
lowing problem. 

48.  To  find  the  ra- 
dius R,  in  terms  of 
the  degree  of  curve 
J).  Draw  OM  perpen- 
dicular to  the  100-foot  chord  AB.  Denote  OA  by  R,  and 
the  angle  AOB  by  D.  Then  in,  the  right  triangle  AMO  we 
have 

50 


R  sin-i-D  =  AM,  whence  R  = 


=  50cosec|D, 


sin  %  D 
and  conversely  the  degree  of  curve  in  terms  of  the  radius. 


(6) 


32  SIMPLE    CURVES. 

Applying  this  formula  to  compute  R  for  a  one  degree  curve 
according  to  the  definition  of  the  degree  of  curve  we  obtain, 

1°  curve,  E  =  5729.65  8°  curve,  R  =  717.78 

2°      "      R  =  2864.93  10°      "      R  =  573.69 

3°      "      12  =  1910.08  14°       "      12  =  410.28 

5°      "      12  =  1146.28  15°      "      12  =  383.06 

7°      "      12=    819.02  20°      "      12  =  287.94 

Comparing  the  corresponding  values  of  12  in  the  two  sets 
given  above  we  perceive  the  difference. 

For  a    1°  curve  =  5729.65  -  5729.58  =  0.07 

"  .2°  "  2864.93-2864.79  =  0.14 

"  3°  "  1910.08-1909.86=0.22 

"  5°  "  1146.28  —  1145.92=0.36 

"  7°  "  819.02-    818.51=0.51 

"  8°  "  716.78-    716.20  =  0.58 

"  10°  "  573.69-    572.96  =  0.73 

"  14°  "  410.28-    409.26  =  1.02 

"  15°  "  383.06-    381.97  =  1.09 

"  20°  "  287.94-    286.48  =  1.46 

The  difference  is  about  a  half  a  foot  in  a  7°  curve  and  about 
a  foot  in  a  14°  curve  ;  so  that  for  ordinary  work  it  is  permis- 
sible to  stake  out  curves  from  1°  to  7°  inclusive  with  chords  of 
100  feet,  using  formula  (5)  to  obtain  12.  From  8°  to  14°  inclu- 
sive use  chords  of  50  feet,  whence 

12  =  — ^—  =  25  cosec  i  D.  (6a) 

sin  £  D 

From  15°  to  28°  inclusive  use  chords  of  25  feet,  whence 

12.5    =  12.5  cosec  £D.  (66) 


sin^D 

For  a  greater  degree,  stakes  should  be  set  in  the  curve  every 
10  feet,  and  the  value  of  the  corresponding  radius 

12  =  -T-^—  =  5  cosec  &  D.  (6c) 

sin  ^L  D 

The  practical  effect  of  the  application  of  the  above  formulas 
is  that  when  the  degree  of  curve  is  assumed,  the  radius  can 
be  determined  at  once  by  simple  division,  and  vice  versa,  if  the 


FUNDAMENTAL    FORMULAS.  33 

radius  is  known,  the  degree  of  curve  Z),  can  be  found.     Thus, 
suppose  D  is  2°. 

Then  E  =  572t9'65  =  2864.83  feet. 

Or,  as  is  frequently  done,  taking  the  radius  of  a  1°  curve  as 
5730  feet, 

£  =  5730  =  2865  feet. 

The  assumption  of  5730  feet  as  the  length  of  the  radius  of  a 
1°  curve  lessens  the  slight  error  committed  in  assuming  the 
equality  of  arc  and  chord  for  the  same  central  angle  and 
radius  when  the  angle  is  2°  and  over  ;  and  the  difference  is 
inconsiderable  for  angles  even  less  than  2°.  That  is  to  say, 

the  radius  of  a  2°  30'  curve  is  more  nearly '—  —  =  2292,  than 

5799  65 
it  is- — riJ_L  =  2291.86,   and  the  radius  of   a  3°  curve  more 

^2 

nearly  one-third  of  5730  =  1910  than  it  is  one-third  of  5729.65 
=  1909.88. 

Tlie  student  may  verify  these  results,  and  explain  why  the 
above  assumption  lessens  the  error. 


EXAMPLES. 

1.  Find  the  length  of  R  when  D  =  6°  38'. 

2.  Find  D  if  7*  =  5000  feet. 

3.  Find  R  it  D  =  30  minutes. 

4.  Find  R   if  D  =  30  degrees. 

5.  Find  the  degree  of  curve  when  R  is  250  feet,  and  state 
how  far  apart  the  stakes  should  be  set. 

49.  Stations  are  usually  placed  100  feet  apart  and  num- 
bered, beginning  at  zero  (0).  Short  chords,  called  sub-chords, 
may,  therefore,  occur  at  the  ends  of  a  curve. 

a.  The  deflection  angle  is  the  angle  IB G  or  GBH,  Fig.  14, 
at  any  point  in  the  curve  subtended  by  a  chord  of  100  feet, 
and  =  i  D. 

b.  The  middle  ordinate  M  is  the  perpendicular  MN  from 
the  middle  of  a  chord  to  the  arc,  or  it  is  that  part  of  the  radius 


34 


SIMPLE    CURVES. 


intercepted  between  the  middle  of  the  chord  and  its  arc,  corre- 
sponding to  the  versed  sine  of  half  the  arc. 

c.  The  long  chord  C,  as  usually  denned,  is  the  line  BE 
joining  the  beginning  and  ending  points  of  the  curve,  though 
the  term  long  chord  is  often  used  in  practice  to  designate  any 
chord  of  the  curve  greater  than  100  feet  in  length,  as  BH. 

d.  The    external    distance  E   is  the   line   Nf  joining  the 
middle  of  the  curve  with  the  intersection  point  7,  and  is,  there- 
fore, the  prolongation  of  the  radius  from  N  to  /. 


o 

FIG.  14. 

50.  The  length  of  a  curve  L  is  given  by  the  number  of 
applications  and  fractions  thereof,  if  any,  of  the  chord  used  in 
laying  out  the  curve,  or  the  number  of  stations  and  fractions 
thereof,  if  any,  which  compose  the  curve. 

In  general  £  =  A. 


The  actual  length  of  arc  = 


300 


FUNDAMENTAL    FORMULAS.  35 

The  distance  from  station  44  to  station  55  =  11  stations  or 
1100  feet.  From  station  44  to  a  point  60  feet  beyond  station 
55  =  1160  feet,  and  such  point  would  be  marked  and  called 
55  _j_  60,  or  may  be  said  to  be  11.6  stations  from  44. 

51.    From  the  properties  of  the  circle  it  will  readily  appear  — 

(a)  That  the  radii  OB  and  OE  are  respectively  perpen- 
dicular to  the  tangents  IB  and  IE. 

(&)    That  the  tangents  IB  and  IE  are  equal. 

(c-)  That  the  angle  of  intersection  a  is  equal  to  the  angle 
BOE  at  the  center,  called  also  the  central  angle. 

(d)  That  the  angle  I  BE  =  IEB  =  $  BOE  =  $  a. 

(e)  That  the  angle  IBN  =  NEI  =  NBE  =  NEB  =  \  a. 
(/)    That  the  angle  IBG  =  £  BOG  =  %D  when  BG  =*  100 

feet. 

When  the  length  of  a  circular  arc  as  BGE  =  the  length  of 
the  radius  BO  or  OE  it  is  called  the  unit  arc,  and  the  angle 
BOE  =  57.o0  ;  hence  if  one  degree  is  subtended  by  a  chord  of 

100  feet 

•   *r  R  =  57.  3  X  100  =  5730  feet. 

Since  the  chords  and  sines  of  small  arcs  or  angles  practi- 
cally coincide,  the  value  of  either  for  1°  being  .01745,  the 
divergence  per  station,  or  100  feet,  will  be  nearly  1.75  feet, 
for  D  =  1°.  For  a  2°  curve,  3.5  feet  ;  and  so  on  in  arith- 
metical ratio,  approximately  correct  for  6°  or  7°,  and  may  be 
used  for  roughly  setting  out  a  curve,  or  as  a  help  and  check 
on  the  instrumental  operations  in  locating  points  in  a  curve. 

EXAMPLES. 

1  .  Given  the  angle  of  intersection  a  =  22°  40'  and  D  =  3°  20', 
to  find  the  length  of  the  curve  : 


3°  20'=    3i°=y>° 

<y  -j-  -y>  =  fi  A  x  -^  =  6.8  stations  =  680  feet.     Ans. 
2.    Given  a  =  31°  51'  and  D  =  2°  46',  to  find  the  length  of 
the  curve  : 

31°  51'  =  1911  minutes 
2°  46'  =    166       " 

=  1  1.51  stations  =  1151  feet.     Ans, 


36  SIMPLE    CURVES. 

REMARK  1.  —  If,  as  in  Example  1,  the  minutes  can  be 
readily  turned  into  convenient  fractional  parts  of  a  degree, 
it  is  best  so  to  reduce  them,  and  then  divide  out.  If, 
however,  as  in  Example  2,  the  fractional  parts  would  not 
be  so  convenient,  it  is  best  to  reduce  both  to  minutes  before 
dividing. 

REMARK  2. —  In  ordinary  preliminary  work,  it  will  be  suffi- 
cient to  measure  lengths  to  the  nearest  foot,  and  to  take  the 
needle-bearings  of  the  tangents.  This  practically  requires  the 
reading  of  the  tape  to  be  made  to  half  a  foot :  that  is,  a  line 
which,  if  read  to  tenths,  would  equal  348.7  feet,  \vould  be 
recorded  349  feet  ;  if  348.4,  would  be  recorded  348.  On  loca- 
tion, however,  the  angles  should  be  read  off  the  plates  by  the 
vernier  to  the  minute,  and  the  measurements  should  be  made 
ordinarily  to  within  two  or  three  tenths.  If  there  be  con- 
ditions requiring  greater  accuracy,  as  in  determining  the  area 
of  valuable  property  in  connection  with  the  right  of  way,  etc., 
then  of  course  the  measurements  should  be  correspondingly 
close,  say  to  a  tenth  of  a  foot.  Xo  strict  rule  can  be  laid 
down  ;  the  engineer  must  use  his  judgment  in  this  matter  as 
in  many  others,  and  make  the  degree  of  precision  consistent 
with  the  interests  involved. 

REMARK  3.  —  For  all  ordinary  calculations  in  the  field  use 
the  natural  functions,  sines,  tangents,  secants,  etc.,  true  to  four 
places  of  decimals,  tables  of  which  are  found  in  this  book. 
Waste  no  time  with  logarithms,  for  even  Without  wind,  rain,  or 
bright  sun  to  contend  with,  the  seeking  of  the  logarithmic 
quantities  in  the  different  tables  —  in  all  field  books  the  figures 
in  the  tables  are  necessarily  small  —  and  then  after  addition 
finding  the  corresponding  number,  will  consume  so  much  time 
that  the  Napierian  follower  will  generally  be  found  far  behind 
one  who  adopts  the  more  primitive  mode  of  computation. 
Table  II,  explained  on  page  42,  containing  tangents  and  ex- 
ternal distances  of  a  1°  curve,  will  also  be  found  useful,  easily 
applied,  and  sufficiently  accurate  for  common  field  practice, 
and  by  consulting  it  for  these  functions  the  transitman  will 
often  save  himself  much  calculation. 


FUNDAMENTAL   FORMULAS. 


37 


3.  Given  the  length  of  a  curve  510  feet,  and  D  =  2°  40'  to 
find  a.     Ans.  13°  36'. 

4.  A  3°  curve  begins  at  station  22  +  36,  and  ends  at  29  +  84, 
find  the  length  of  the  curve, 

the  radius,  and  central  angle. 

5.  Given  the  angle  of  inter- 
section 53    57',  and  the  degree 
5°  43'  to   find   the  length  of 
the  curve. 

6.  How  much  longer  is  the 
arc  than  its    100-foot   chord, 
the  radius  being  500  feet  ?     If 
the  radius  is  1000  feet  ?  2000 
feet? 

52.  Given  the  angle  of 
intersection  a,  and  the  radius 
R,  or  degree  D,  to  find  the 
tangent  distance. 

Ir*  the  right  triangle  BOI  the  angle  BOI=±  a. 

Therefore  r=/?tan|a,  (7) 

and  substituting  for  R  its  value  from  (5) 


This  formula  is  useful  when  the  radius  OB  or  degree  of  curve 
is  assumed,  the  tangent  distance  IB  must  then  be  calculated. 
If,  on  the  other  hand,  T  be  assumed,  then  we  may  find  R,  and 
hence  Z),  from  the  problem  — 

53.  Given  the  tangent  distance  T  and  the  central  angle  a, 
to  find  the  radius. 

From  (7) 

R=Tcot±a.  (9) 


EXAMPLES. 

1.    Given  the  tangent  distance  450  feet,  and  the  angle  of 
intersection  23°  42'  to  find  the  radius. 


38  SIMPLE   CURVES. 

2.    Given  the  angle  of  intersection  17°  56',  and  the  degree  of 
curve  2°  40',  to  find  the  tangent  distance. 

54.    Given  the  radius  R  and  central  angle  a,  to  find  the 
long  chord  C. 

In  the  right  triangle  BOm  Fig.  16 

Bm  =  BOsmBOm 

C 
or,  —  =  Rsm$a. 

Hence  C  =  2Rsm$a.  (10) 

The  student  may  show  that, 


(11) 

and  that  C  =  2  M  cot*  or.  (12) 

Example  —  Given  a  2°  50'  curve,  having  a  central  angle  of 
28°  30',  to  find  the  long  chord. 

55.  Given  the  radius  R  and  central  angle  a,  to  find  the 
middle  ordinate  M. 

mN=  BO  versine  BOm-, 
or,  M  =  R  versine  £  a.  (13) 

Example  —  Given  a  3°  40'  curve  having  a  central  angle  of 
32°  50'  to  find  the  middle  ordinate. 

If  a  and  the  external  distance  E  are  given,  to  find  M, 
substitute  in  (13)  the  value  of  R  from  the  proportion, 

cosine  :  R  =  versine  :  exsec,  or  JE",  and  we  obtain, 

M=Ecos$a.  (14) 

The  student  may  verify  the  last  equation  by  another 
method. 

56.  Given  the  radius  R  and  chord  C  to  find  the  middle 
ordinate  M. 

mN  =  ON  —  Om. 

But  Om  =  *J OE"  —  mE2> 

and  substituting  values,  there  results, 

•?•  (15> 


FUNDAMENTAL   FORMULAS. 


39 


57.  Given  the  radius  and  chord,  to  find  any  ordinate  of  a 
curve,  its  distance  from  the  center  of  chord  being  known. 

In  Fig*.  16  let  ap  be  the  ordinate  whose  length  is  required. 
Extend  pa  to  Q,  draw  OQ,  parallel  to  the  chord  BE,  and  join 
Op.  Denote  the  distance  am  by  f/,  and  the  other  notation  as 
above;  then  in  the  right  triangle  p 0Q, 


But 


and 


substituting  these  values,  there  results, 


(16) 

If  d  =  0,  (16)  reduces  to  (15),  as  it  evidently  should. 

For  most  practical  purposes  a  modification  of  formula  (15) 
may  be  employed  to  locate  points  in  a  curve. 

By  expanding  the  binomial  of  the  right-hand  member  in  a 
series  to  three  terms,  there  results, 


40  SIMPLE    CURVES. 

The  last  term  of  this  series  will  not  affect  the  result  .03  of  a 
foot  for  M  of  a  chord  100  feet  and  radius  300  feet,  and  may 
be,  therefore,  safely  rejected;  we  have  then, 

*=g;  (IT) 

And  hence  for  any  other  middle  ordinate  Ml  and  chord  Cl 
in  the  same  curve, 


.  (17a) 

SB 


or,  M:Ml=C*: 


If  Ci  =  iC  Mi  =  ¥.  (18) 

Assuming  BN  —  %  BE  (which  may  often  be  done  with  suf- 
ficient accuracy),  its  middle  ordinate  m'n'  =  ±  M,  and  the  mid- 
dle ordinate  of  En'  =  ±  m'n',  and  so  on,  numerous  points  in  a 
curve  may  be  established. 

Practically  these  points  may  be  located  by  measuring  off  the 
computed  distance,  as  m'n',  from  the  middle  point  of  a  tape 
stretched  between  the  extremities  of  an  arc,  as  BN. 

EXAMPLES. 

1.  Find  the  middle  ordinate,  and  the  length  of  one  25  feet 
therefrom,  of  a  100-foot  chord,  the  radius  being  1000  feet. 

2.  Given  the  radius  of  a  curve  500'  and  chord  100'  to  locate 
in  the  arc  points  whose  projections  on  the  chord  shall  be   12^ 
feet  apart.     Compare  results  by  different  methods. 

58.  Given  the  radius  R,  and  central  angle  a,  to  find  the 
external  distance  E.  As  Bin  is  the  sine  and  mN  the  versed 
sine  to  radius  R  of  the  arc  BON,  Fig.  17,  so  is  NI  the  external 
secant  of  the  same  arc  and  radius. 

Hence  given  the  external  distance  E  and  central  angle  a 
to  find  the  radius  R, 


R  = (20) 

exsec^a 


FUNDAMENTAL   FORMULAS. 


41 


The  student  may  show  that 

E=  T cottar.  exsec-Jar, 
and  that  E  =  M  sec  £  a. 


(21) 
(22) 


59.    Given  the  tangent  distance  T,  and  the  central  angle 
a,  to  find  the  external  distance  E. 

In  Fig.  17  extend  the  radius  OB  until  it  meets  at  K,  the 
tangent  drawn  from  the 
point   N  of   the 
Also  extend  ON 


middle 

curve. 

until  it  meets,  at  /,  the 

tangent  drawn  from  the 

extremity  of  the  curve  at 

E.     Join  IK,    and  BN. 

By   this   construction    it 

is  evident   that  IK  and 

BN    are    parallel ;    that 

NK  =  BI;    that   NT   is 

the    external    secant    of 

the   arc   BON,   and    the 

angle    IKN  =  EBN  =  $ 

BON  =  \  a,  and  therefore  in  the  right  triangle  NIK 

NI  =  NKtsuiNKI, 

or,  E=Tta,n±a.  (23) 

Hence  given  the  external  distance  E  and  central  angle  a 
to  find  T,  we  obtain  from  (23) 

(24) 


The  student  may  show  that 


E 


M 


=  M  seeder. 


(25) 


REMARK.  —  Equations  (20)  and  (24)  will  aid  us  in  deter- 
mining the  elements  and  retracing  the  curved  track  of  a  rail- 
road when  all  notes  concerning  it  are  defaced,  and  the  location 
of  the  P.C.  and  P.T.  are  unknown.  Extend  the  centre  lines 
of  the  tangents  to  the  curve,  to  their  intersection  at  /,  Fig.  17  ; 
observe  the  angle  supplementary  to  a,  bisect  it,  and  measure 


42  SIMPLE   CURVES. 

on  the  bisector  from  I  to  the  center  of  the  track,  and  thereby 
get  E.  The  values  of  E  and  a,  being  thus  discovered  and 
substituted  in  the  equations  named,  make  known  R  and  T. 

EXAMPLES. 

1.  Given  the  central  angle  28°  48',  and  degree  of  curve  4°  40', 
to  find  the  external  distance. 

2.  Given  the  angle  of  intersection  44°  56',  and  the  tangent 
distance  560  feet,  to  find  how  far  from  /  to  the  nearest  point 
on  the  curve. 

3.  The    angle   of    intersection   is   32°  20',    and   the   curve 
passes  within  60  feet  of  /.     Find  the  distance  from  the  P.I. 
to  the  P.O. 

4.  The    angle  of   intersection   is  26°  40',  and  the  nearest 
point  of  the  curve  is  to  be  56  feet  from  /.     Find  the  radius 
and  degree  of  curve. 

Tangents  and  external  distances  for  a  one-degree  curve  for 
every  ten  minutes  of  central  angle  are  arranged  in  table  II. 
In  this  table,  therefore,  one  may  find  the  length  of  tangent  or 
external  distance  at  once  corresponding  to  a  radius  of  5730 
feet,  and  a  central  angle  varying  by  ten  minutes  of  arc,  and 
by  interpolation  he  may  obtain  these  lengths  for  any  minute 
whatever. 

For  the  required  length  of  a  corresponding  tangent,  or 
external  distance  of  any  other  radius  or  degree  of  curve,  take 
the  proportional  part  thus:  To  find  the  length  of  a  tangent 
corresponding  to  a  3°  curve  and  central  angle  24°,  look  in 
table  II  under  24°,  and  take  out  of  the  column  of  tangents, 
opposite  24°,  the  number  1218.  This  is  the  length  of  the  tan- 
gent of  a  1°  curve  having  a  central  angle  of  24°. 

Now  the  tangent  of  a  3°  curve,  and  the  same  central  angle, 
is  only  one-third  as  long;  hence,  1218  -f  3  =406  =  length  of 
chord  required. 

The  student  may  verify  this  result  by  either  formula  (7) 
or  (8). 


FUNDAMENTAL   FORMULAS.  43 

EXAMPLES. 

[To  be  solved  by  aid  of  Table  II.] 

1.  Given  the  intersection  angle  a  =  48°  26',  and  D  =  5°,  to 
find  the  tangent  distance. 

2.  Given  a  =  28°  20',  and  the  length  of  the  tangent  =  361.6 
feet,  to  find  the  radius.  Ans.  1432.5  feet. 

3.  Given  the  angle  a  ==  36°  40',  to  find  the  external  distance 
and  tangent  of  a  3°  40'  curve. 

4.  Given  the  tangent  distance  T  =  559  feet,  and  D  =  4°,  to 
find  a.  Ans.  42°  38'. 

5.  Given  the  external  distance  126.1  feet,  and  D  =  2°  40', 
to  find  a. 

SOLUTION.  —  Find  the  product  of  126.1  by  2f  =  336.3  in 
the  table  in  the  E  column,  and  note  the  degrees  and  minutes 
corresponding  thereto  =  38°  20.  Ans. 

The  following  formulas  derived  on  preceding  pages  are 
grouped  here  for  convenience  of  reference. 

E  =  Radius.  C  =  Long  chord. 

L  =  Length  of  curve.  M  =  Middle  ordinate. 

T  =  Tangent  distance.  E  =  External  distance. 

c  =  Any  chord. 


sin^D  2 


R  =  -  —  -  T  = 


= 


M  = 


exsec^a: 
C 


versine  4-  a 
L  =  100  - 


SIMPLE   CURVES. 


=-      (nearly) 


E=  Ttan^o: 


E  =  M  sec  \  a 


O 

FIG.  18. 

B.    LOCATING   SIMPLE    CURVES. 

60.  Given  the  degree  D  to  locate  a  curve  from  a  known 
point  in  a  given  tangent. 

Let  B  in  the  above  figure  be  the  known  point  in  the  tangent 
AL  Set  up  the  transit  at  5,  level,  and  for  convenience  make 
the  zeros  of  the  plates  coincide.  Without  disturbing  the  rela- 
tive positions  of  the  zeros,  observe  some  point,  as  /  or  J  in  the 
tangent  for  direction;  then  turn  off  the  angle  IBC  =  ^D,  and 


LOCATING    SIMPLE   CURVES.  45 

measure  in  the  direction  of  the  line  of  sight  100  feet  and  there 
set  C.  Deflect  again  CBF=  £Z>,  i.e.,  make  the  reading  of  the 
plates  now  =Z),  and  measure  from  C  onward  100  feet,  to  a 
point  F  in  the  line  of  sight,  thus  locating  F.  Deflect  again 
the  angle  FBG  =  £D,  making  the  reading  of  the  instrument 
| D,  and  measure  100  feet  from  F  to  G,  and  so  on  as  far  as 
may  be  necessary.  If  the  curve  is  to  be  extended  farther  than 
it  can  be  seen  from  the  point  B,  the  direction  of  the  tangent 
at  the  point  011  the  curve  on  which  it  is  desired  to  place  the 
instrument  must  be  ascertained;  hence  the  problem. 

61.  To  find  the  direction  of  a  tangent  at  a  given  point  in 
a  curve.  Let  G  be  the  point,  and  GP  the  tangent.  The  read- 
ing of  the  plates  when  sighting  G  was  |Z>;  clamp  at  that  read- 
ing, and  transfer  the  instrument  to  G,  see  that  the  index  is  not 
disturbed,  and  sighl  to  B,  then  turn  off  an  angle  BGK=  GBK, 
i.e.,  =  -|Z),*  or  make  the  reading  of  the  instrument  3Z),  and  the 
telescope  will  point  in  the  direction  of  the  required  tangent 
GK,  whence  inverting  the  telescope  and  making  proper  de- 
flections as  before,  other  points  in  the  curve  may  be  found. 
Having  run  the  curve  from  this  new  tangent  point  G,  two 
more  stations  to  E,  let  us  suppose  it  is  desired  to  turn  into 
tangent  at  E.  Clamp  the  index  at  its  last  reading,  that  is  4D, 
set  up  at  E,  and  with  the  index  undisturbed  observe  G,  then 
turn  off  an  angle  GEP  =  EGP,  the  angle  deflected  from  the 
tangent  at  the  last  station,  and  the  telescope  will  point  in  the 
direction  of  the  required  tangent. 

The  following  is  a  general  rule  for  this  common  operation  : 

For  direction  of  tangent  at  any  point  in  a  circular  curve : 
—  From  twice  the  reading  of  the  instrument  when  locating  the  point, 
subtract  its  reading  at  the  last  tangent,  or  as  we  sometimes  say 
for  shortness,  double  the  index  minus  the  last  tangent. 

To  illustrate  further  :  — 

Suppose  the  reading  of  the  instrument  at  G,  when  the  tele- 
scope pointed  in  the  direction  of  GK,  to  be  12°,  the  degree  of 
curve  being  4°  ;  then  when  sighting  E,  the  reading  or  index 
will  be  16°.  Now  after  making  at  E  the  observation  on  G  as 

*  The  student  will  perceive  that  the  triangle  BGK  is  isosceles. 


46  SIMPLE   CURVES. 

directed  above,  turn  into  tangent  by  setting  the  index  at  2  x 
16°  — 12°  =  20°,  and  having  started  at  B  with  the  index  at 
zero,  the  reading  20°  indicates  the  magnitude  of  the  central 
angle  BOE,  or  the  angle  of  intersection  a. 

The  student  may  verify  by  summing  the  deflections. 

REMARK!.  —  It  will  be  perceived  that  at  any  tangent  point 
when  the  telescope  points  in  the  direction  of  the  tangent  the 
reading  of  the  vernier  gives  exactly  the  amount  of  the  central 
angle  consumed. 

Hence,  when  the  angle  of  intersection,  or,  of  two  tangents  is 
measured,  the  P.C.  and  P.T.  located,  and  then  a  curve  traced 
uniting  these  points,  a  check  on  the  work  is  secured,  for  at  the 
P.T.  when  the  telescope  points  in  the  direction  of  the  located 
tangent,  the  reading  of  the  plates  should  be  a°. 

REMARK  2.  —  Some  engineers  before  taking  the  back  sight 
from  the  new  tangent  point,  turn  back  the  zero  of  the  vernier 
past  the  zero  of  the  limb,  just  as  far  as  it  was  on  the  other 
side  when  the  new  tangent  point  was  sighted.  Then  after 
sighting  the  previous  tangent  point,  move  the  vernier  plate 
back  to  zero,  thus  bringing  the  telescope  to  point  in  the  direc- 
tion of  the  new  tangent.  But  this  is  objectionable,  since  it 
always  requires  two  changes  and  two  readings  of  the  plates, 
takes  more  time,  it  is  obviously  no  more  accurate,  and  more- 
over renders  impossible  the  convenient  check  which  the  more 
expeditious  method  introduces. 

The  following  are  the  field  notes  of  a  3°  curve,  central  angle 
25°  30',  P.C.  at  station  24,  curve  turned  to  the  right. 

The  student  should  calculate  the  tangent  distance,  or  find  its 
length  432  feet  from  Table  IT.  He  should  also  compute  the 
length  of  the  curve  =  8£  stations  — 850  feet;  determine  the 
deflection  angle,  and  the  amount  to  deflect  at  the  P.T.  for  the 
sub  chord ;  the  reading  of  the  vernier  at  the  tangent  points 
when  telescope  is  pointing  in  direction  of  tangent ;  and  in  fact 
he  should  verify  the  work  in  every  particular. 


LOCATING   SIMPLE   CURVES. 


47 


Sta- 
tion. 

De- 
flect. 

Read- 
ing. 

Tan- 
gent. 

C'mputed 
Course. 

Magnetic 
Course. 

Remarks. 

P.T. 

32+50 

0°45' 

21°  45' 

25°  30' 

N75°20'E 

N75°30'E 

32 

21°  00' 

31 

19°  30' 

T.P. 

30 

13°  30' 

18°  00 

29 

12°  00' 

28 

10°  30' 

T.P. 

27 

4°  30' 

9°  00' 

26 

3°  00' 

25 

1°30' 

1°30' 

P.C. 

24 

0 

0 

3°  curve,  turning  right, 

a  =  25°  30'     T-  432.2  feet. 

Make  a  complete  table,  as  above,  for  the  following 
problems  — 

1.  A30  40'  curve,  turning  right,  central  angle  48°  40',  P.C. 
at  station  51  -f-  60.     Make  three  tangent  points  between  the 
P.C.  and  the  P.T. 

2.  A  9°  30'  curve  to  the  right,  a  =  70°  P.C.  at  station  84, 
make  two  T.P.'s. 

Another  method,  in  which  the  deflections  for  all  proposed 
stations  may  be  calculated  before  going  on  the  field.  The 
record  is  shown  in  the  table  below. 


P.T. 

T.P. 
T.P. 

P.C. 


Sta- 
tion. 

Deflec- 
tion. 

Read- 
ing. 

C'mputed 
Course. 

Magnetic 
Course. 

Remarks. 

+  50 

0°40' 

10°  00' 

17 

9°  20' 

16 

8°  00' 

15 

6°  40' 

14 

5°  20' 

13 

4°  00' 

12 

2°  40' 

11 

I8  20' 

1°20' 

10 

0 

0 

2°  40  curve,  turn'g  right, 
a  =  20°        T=  378.9 

Set  up  the  transit  at  station  10,  the  P.C.,  and  proceed  as  in 
the  previous  case  to  locate  11,  12,  and  13.  Carry  the  transit 
to  13,  clamp  the  zeros  together,  and  then  observe  the  P.C. 
Then,  if  the  vernier -plate  is  moved  over  4°  in  the  direction  of 
the  curve,  the  telescope  will  point  in  the  line  of  tangent  at 
station  13  ;  and,  with  the  index  at  5°  20'  (the  reading  oppo- 


48  SIMPLE    CURVES. 

site  14),  station  14  may  be  set,  and  6°  40'  will  locate  15. 
Then  transfer  the  instrument  to  15,  make  the  reading  4°  (the 
deflection  corresponding  to  13,  the  last  T.P.),  and  observe  13  ; 
clamp  the  lower  plate.  Now,  when  the  vernier  reads  6°  40', 
the  telescope  will  point  in  the  direction  of  the  tangent  at 
station  15,  and  readings  of  8°,  9°  20',  and  10°,  respectively,  will 
set  the  remaining  stations  and  sight  the  P.T.  If  now  the 
instrument  be  set  up  at  the  P.T.  and  an  observation  made 
on  station  15  with  the  vernier  at  6°  40',  the  lower  plate  then 
clamped,  and  the  reading  made  10°,  the  telescope  will  point  in 
the  direction  of  the  tangent,  and  this  reading  is  evidently  one 
half  the  central  angle. 

REMARK.  —  Observing  T.P.  13  from  T.P.  15  with  the  index 
at  4°,  turning  into  tangent  at  6°  40',  and  sighting  station  16 
with  the  vernier  at  8°  is  evidently  the  same  as  though  the  P.O. 
were  sighted  from  15  with  the  plates  at  zero,  the  tangent  turned 
at  6°  40'  and  station  16  set  with  the  index  at  8°.  Before 
making  an  observation  at  any  station  set  the  vernier  at  the 
reading  opposite  the  point  on  which  the  observation  is  to  be 
made,  observe  the  point,  clamp  the  lower  plate,  then  any  station 
in  the  curve,  either  way  from  the  instrument,  may  be  located 
by  setting  the  vernier  at  the  reading  opposite  that  station, 
and  the  telescope  will  point  the  direction  of  tangent  at  the 
station  when  the  index  points  to  the  reading  opposite  the 
station. 

The  advantages  which  this  method  possesses  over  all  others 
are  that  the  deflections  required  for  all  proposed  stations  or 
known  chord  lengths  may  be  tabulated  in  advance,  and  the 
deflections  may  be  used  to  run  the  curve  in  from  either  end,  or 
from  any  intermediate  point,  working  either  way  from  the 
instrument.  Of  course  any  odd  plus  employed  to  locate  a 
point  not  predetermined  will  have  to  be  calculated  on  the 
field. 

62.  To  locate  points  in  a  curve  of  given  radius  by  off- 
sets from  a  given  tangent. 

Let  B  represent  the  beginning  point ;  c',  e',  f,  points  in  the 
tangent  to  be  found  in  order  to  establish  c,  e,f,  points  100  feet 


LOCATING    SIMPLE    CURVES. 


49 


apart  in  the  curve.  The  distance  from  B  along  the  tangent 
to  the  points,  and  the  lengths  of  the  offsets  are  required.  From 
the  known  radius  the  angle 
BOc  =  D  is  found,  and  it 
it  is  evident  that 


FIG.  19. 


Be'  =  m'e  =  R  sin  2  D  ; 
and    similarly    for    other 
points. 
Also 

c'c  =  Bin  =  E  versine  D  ; 

e'e  =  Bin'  =  R  versine  2  D  ; 
and  so  on  for  other  points. 
From  these  equations 
the  distances  and  offsets 
may  be  computed,  and 
then  measured  off,  or  they  may  be  taken  at  once  from 
tables  of  middle  ordinates  and  long  chords,  since  it  will 
be  perceived  that  the  offsets  Bm,  Bm',  etc.,  correspond  to  middle 
ordinates  of  double  the  arcs  Be,  Be,  etc.,  and  me,  m'e,  etc.,  cor- 
respond to  one-half  the  long  chords  of  double  the  arcs.  If 
there  is  a  sub-chord  at  B  the  above  equations  will  still  give 
the  proper  distances  by  substituting  for  D  the  value  of  the 
angle,  say  8,  which  the  sub-chord  subtends,  or 

*  R  sin  8,  and  for  the  next  point 
R  sin  (8  +  D),  and  so  on. 

The  same  substitutions  will  be  required  in  the  equations 
containing  the  versines. 

EXAMPLE.  —  Find  the  distances  along  the  tangent  and  the 
lengths  of  the  offsets  for  a  curve  of  2000  feet  radius. 

63.  To  locate  a  curve  of  given  radius  by  offsets  from 
chords  produced. 

Produce  the  chord  ac  to  e',  Fig.  20,  draw  bch  tangent  to  the 
curve  at  c,  and  draw  ee'  through  h  perpendicular  to  ch,  and 
suppose  a,  c,  e,  and  y  points  in  the  curve.  Then  since  the  angle 
e'ch  =  angle  ech,  and  the  side  ch  perpendicular  to  ee',  these  tri- 


50 


SIMPLE    CURVES. 


angles  are  equal ;  the  side  ce'  =  ce  and  the  triangle  cee'  is 
isosceles.  Moreover,  since  the  angle  ecef  =  cOe  and  both  tri- 
angles are  isosceles,  they  are  similar  and  the  corresponding- 
sides  are  proportional ;  hence, 

cO :  ce  =  ce:  ee', 
or  E  :  C    =  C  :  ee' ; 

.-.  ee'  =  <¥-.  (26) 


If  C  =  100,  as  it  does  usually,  then 

1002 
R 

£ 


ee'  = 


(2Ca) 


FIG.  20. 

Otherwise  from  the  equality  of  the  triangles  ceh  and  ce'h,  it 
will  be  readily  perceived  that  the  chord  offset  ee'  =  twice  the 
tangenf  offset  he,  and  therefore  the  equations  for  tangent 
offsets,  last  articles  are  applicable.  Practically  each  chord  as 
ac  must  be  extended  through  c  to  e'  a  distance  of  100  feet,  and 
then,  while  preserving  the  point  e',  draw  the  tape  over  from  e' 
past  h  keeping  it  pivoted  at  c  until  the  point  e  is  found,  which 
shall  be  100  feet  from  c  and  the  computed  distance  from  e'. 

If  there  is  a  sub-chord  at  the  beginning,  its  complement,  or 
what  it  lacks  of  100  feet,  may  be  laid  off  in  the  opposite  direc- 
tion by  offsets  from  the  tangent,  and  then  having  the  extremi- 


OBSTACLES. 


51 


ties  of  the  chord  proceed  as  above.  In  either  case  if  the  chord 
is  not  given  in  this  direction  find  it  by  offset  from  tangent,  as 
in  article  62. 

EXAMPLE.  —  Given  the  sub-chord  of  30  feet,  located  in  a 
curve  whose  radius  =  1910  feet,  to  find  the  necessary  deflections 
and  lay  out  the  curve. 

C.    OBSTACLES. 
64.    To  pass  an  obstacle  on  a  curve  : 

Let  0  be  the  center  of  the  curve  y  x,  which  we  may  suppose 
is  laid  out  in  the  usual  manner  by  deflection  angles,  until  the 
point  p  is  reached,  when  obstacles  shown  in  the  figure  inter- 
vene, requiring  evidently  some  modification  or  change  in  the 
method  being  employed. 


FIG.  21. 

I  The  best  way  ordinarily  is  to  turn  off  an  angle  equal  to  a 
certain  number  of  deflection  angles  to  just  pass  the  obstacle. 
In  the  present  case  four  times  one  deflection  will  do  ;  compute 
the  long  chord,  as  pt,  or  take  its  length  from  table  Y.  Meas- 
ure out  this  distance  to  t,  set  up  there  ;  turn  into  tangent,  make 
the  proper  deflection,  and  if  possible  set  some  of  the  back 
stations,  as  s  and  r,  and  then  before  lifting  the  instrument 
locate  other  points  if  possible  between  t  and  x.  Points  q  and 
r  may  be  located  by  offsets  from  the  tangent  pv,  Art.  62. 


52 


SIMPLE   CURVES. 


65.  To  locate  a  curve  when  the  point  of  intersection  is 
inaccessible. 

When  the  P.T.  is  inaccessible  neither  the  tangent  distance 
nor  the  angle  a  can  be  measured  directly,  hence,  some  equiva- 
lent must  be  discovered. 

The  tangents  being  lo- 
cated as  near  as  practicable 
to  the  P. I.,  set  up  at  some 
point  I  in  the  one,  and  ob- 
serve some  point  n  in  the 
other;  note  the  angle  nil, 
measure  In,  and  also  the 
angle  Inl.  The  sum  of  the 
two  angles  measured  =  cr. 
From  the  data  obtained  com- 
pute the  distance  II ;  meas- 
ure from  /  towards  p,  a  dis- 
tance =  to  the  difference  be- 
tween the  assumed  tangent  distance  and  77,  thus  locating  the 
P.C.,  whence  the  curve  may  be  traced  as  usual. 

If  it  is  impossible  to  run  a  straight  line,  survey  a  broken 
line,  and,  as  in  working  a  traverse,  determine  the  direction 
and  length  of  In ;  then  proceed,  as  before,  to  find  the  P.C. 

66.  To  locate  a  curve  when  the  point  of  curve  is  inac- 
cessible. 

Let  I  be  a  numbered  station  at  the  point  of  intersection  of 
two  tangents  al  and  It,  Fig.  23;  73,  the  P.C,  the  distance  to  which 
from  /  being  known.  Then  if  there  are  two  or  more  stations  in 
the  vicinity  of  a  and  either  of  them  numbered,  the  location  of 
any  point  as  c'  is  discovered;  a  tangent  offset  therefrom  will  fix 
a  point  c  in  the  curve  extended  back  of  the  P.C.  Set  up  at  c, 
and  deflect  from  cc'  a  right  angle  less  B  Oc,  the  telescope  will 
then  point  in  the  direction  of  tangent  at  c.  Then  deflect  suf- 
ficiently to  clear  the  obstacle,  usually  to  some  station,  measure 
the  long  chord  eg,  and  obtain  thereby  a  point  in  the  curve 
beyond  73,  whence  the  curve  may  be  traced  in  the  usual  man- 
ner. If  the  preceding  is  impracticable,  set  up  at  some  point 


OBSTACLES. 


53 


/';  its  position  may  be  known  by  the  stakes  in  tangent  or  its 
distance  from  /  may  be  measured.     Compute,  and  observe  the 


FIG.  23. 


angle  a:',  and  measure  the  distance  I'n  =  FB.  Move  to  n,  sight 
/',  the  telescope  will  then  point  in  the  direction  of  the  tangent, 
and  deflections  may  be  made  either  way  to  set  points  in  the 


67.    To  locate  a  curve  when   both  the  point   of   curve 
and  point  of  intersection  are  inaccessible. 


o 
FIG.  24. 


Let  a  I  and  It  represent  the  tangents.     Set  up  at  some  point 
I  in  the  tangent  al ;  run  a  line  In,  joining  the  tangents,  as  in 


54 


SIMPLE   CURVES. 


Article  65  ;  measure  the  angles  at  I  and  n,  and  compute  the 
distance  II.  Then,  knowing  the  radius  and  central  angle, 
compute  IB,  and  thence  Bl  becomes  known.  Now,  by  tan- 
gent offset,  as  in  the  previous  article,  a  point  c  in  the  curve 
may  be  located  ;  place  the  transit  there,  make  proper  deflec- 
tion to  clear  the  obstacle,  obtain  the  corresponding  long  chord 
ce,  and  proceed  to  completely  trace  the  curve,  as  in  the  pre- 
ceding article. 

68.  To  pass  from  curve  to  tangent  when  the  point  of 
tangent  is  inaccessible. 

Let  tt'  be  the  tangent  sought 
and  t,  the  inaccessible  P.T. 
Locate  as  many  stations  in 
the  curve  as  practicable;  then 
deflect  at  some  point  n,  so  as 
to  clear  the  obstacle  and  set 
a  point  c  in  the  extension  of 
the  curve  beyond  the  P.T. 
^  by  the  long  chord  as  in 
previous  articles.  Compute 
the  tangent  offset  cc'  =  R 
versine  cOt ;  measure  it  off, 
making  the  angle  ncc'  =  a 
right  angle  plus  the  angle 
which  the  long  chord  makes 
with  the  tangent,  that  is  90° 
-f  ncrc',  (n'c  being  drawn 
parallel  to  the  tangent).  A 

right  angle  turned  at  c'  with  c'c  will  give  the  direction  of 
tangent,  and  the  distance  c't  =  R  sin  cOt. 

If  c  is  farther  than  n  from  t,  the  angle  c'cn  will  equal  90° 
minus  the  angle  which  the  chord  makes  with  the  tangent. 
In  general  c'cn  =  90°  ±  ncn'. 

If  the  deflection  for  long  chord  at  n  is  made  for  a  certain 
number  of  stations,  the  length  of  the  chord  may  be  taken 
direct  from  Table  V. 

Or  compute  the  length  of  the  tangent  offset  at  the  known 


FIG.  25. 


OBSTACLES. 


55 


point  n,  and  sight  from  n  and  measure  direct  to  some  point 
c"  on  the  tangent,  its  distance  being 


nc"  = 


nn 


sin  nc"n" 

Then  in  the  triangle  nc"n"  calculate  c"n"  and  substract  from 
it  tn"  =  11  sin.  n0f,  thus  obtaining  c'%  whence  the  numbering 
of  the  stations  may  be  properly  continued.  If  an  angle 
tnc"  be  turned  at  n  and  made  one-half  ntn"  =  one-fourth  nOt, 
ic"  will  equal  tn",  and  no  calculation,  except  for  nc",  will  be 
necessary. 

If  the  angle  at  c"  is  very  acute  this  method  will  not  be  very 
reliable. 

Otherwise,  if  the  obstruction  does  not  prevent  alignment. 


FIG.  26. 

From  a  known  point  n,  in  the  curve,  Fig.  26,  compute  the 
tangent  distance  ni ;  measure  it  oif,  locate  i  and  set  the  instru- 
ment there.  Deflect  a'  from  ni,  plunge  the  telescope  and  set  a 
point  e  in  tangent ;  sight  another  point  c  along  the  bank  of 
the  stream,  measure  the  angle  eic  and  the  length  of  ec.  Then 
in  the  triangle  cie  compute  ie,  subtract  ti  =  ni  from  it  and 
thereby  obtain  te. 


56 


SIMPLE   CURVES. 


69.    To  extend  a  curve  across  a  pond  or  stream. 

Let  pt  be  the  tangent  to  a  curve  pv  at  the  point  p.  Set  a 
point  c  in  the  tangent,  and  estimate  the  distance  to  a  point 
n  across  the  water,  using  a  certain  number  of  stations,  and 
deflect  accordingly  the  angle  tpn.  Move  to  n,  measure  the 
angle  pnc  and  the  distance  nc.  Then  by  the  triangle  pnc  com- 
pute pn,  compare  the  computed  and  estimated  distances,  and 


FIG.  27. 

discover  thereby  the  precise  position  of  n.  If  it  is  not  on  the 
curve  (it  probably  will  not  be),  measure  from  n  to  n',  the 
difference  between  the  computed  and  estimated  distances  of_/m, 
either  forward  or  backward,  as  the  case  may  require.  Place 
the  transit  at  n',  turn  into  tangent,  and  proceed  in  the  usual 
manner  to  set  other  points  in  the  curve. 

D.   PROBLEMS   IN   CHANGE    OF   LOCATION. 

70.  Having  located  a  curve  between  two  tangents,  it  is 
required  to  determine  the  necessary  change  in  the  radius, 
and  the  external  distance  for  any  desired  change  in  the  tan- 
gent distance. 

Denote  BI  =  IN  by  T. 

B'I  =  IN'\>y  T'. 
«        OB  =  ON  by  R. 
«      O'B'  =  &N'  by  R'. 
"      Jm  by  E. 
«*      Im'  by  E'. 


PROBLEMS  IN  CHANGE  OF  LOCATION.     57 

Then  BE'  =T  —  T'  =  the  given  change  in  tangent  distance. 
Draw  O'D  perpendicular  to  OB,  it  will  equal  T—  T,  and  OD 


Then  in  the  right  triangle  ODO', 

R  —  R'  =  (T  —  T)  cot.  i  a.  (27) 

From  (23)  E  =  T  tang.  £  a. 

E'  =  T  tang.  \  a. 
.:E-E'  =  (T-  T)  tang.  £  a.  (27a) 

EXAMPLES. 

1.  Two  tangents  which  intersect  at  an  angle  of  32°  are 
united  by  a  3°  30'  curve.  It  is  desired  to  lengthen  the  tan- 
gent distance  72  feet;  find  the  necessary  change  in  R  and  E. 

By  (27)  E  —  E'  =  72  cot  16°, 

JJ-E/=72  X  3.2709  =  237.5; 
.-.  R  =  1637.28  +  237.5  =  1874. 78, 
or  a  curve  of  about  3°  3'. 

By  (27a)  E-E'=72  tan  8°, 

or  E-E'  =  72  X  0.1405  =  10.12. 


58  SIMPLE    CURVES. 

2.  Find  the  change  in  the  degree  of  curve  in  Example  1,  on 
the  assumption  that  T  is  to  be  shortened  60  feet. 

71.  Having  located  a  curve  between  two  tangents  it  is 
required  to  determine  the  necessary  change  in  the  radius 
and  tangent  distance,  for  any  desired  change  in  the  external 
distance.  (Fig.  28.) 

By  (19)  mI=E  =  R  exsec  |  a-, 

m'l  =  E'  =  R'  exsec ± a; 

or,  R-R'  =  E~E/  (28) 

exsec  i- a 

By  (27a)  T-  T'  =  (E-  E')  cot* a.  (29) 

EXAMPLE.  —  Two  tangents  which  intersect  at  an  angle  of 
40°  are  united  by  a  3°  40'  curve.  It  is  desired  to  bring  the 
middle  point  of  the  curve  30  feet  nearer  the  P.I.  Find  the 
length  of  Ef  and  T. 

30 


By  (28)  R-R'  = 


exsec  20 


jl-jl'  =  _       =  407.29. 
.0642 

...  E'  =  1562.88  -  467.29  =  1095.59, 
or  a  curve  of  5°  14' 

By  (29)  T  —  T  =  30  cot  10, 

T  -  T  =  30  X  5.6713  =  170.14. 

...  T  =  568.84  -  170.14  =  398.7. 

72.  Having  located  a  curve  between  two  tangents  it  is 
required  to  determine  the  necessary  change  in  the  tangent 
distance  and  external  distance  for  any  desired  change  in  the 
radius. 

From  (27)  we  obtain 

T  —  T  =  (R  —  R')  tan  £  a.  (30) 

From  (28)        E-E'  =  (R-  R')  exsec  |  a.  (31 ) 

EXAMPLE.  —  Two  tangents  which  intersect  at  an  angle  of 
28°  40'  are  united  by  a  4°  20'  curve.  Find  what  change  would 
be  made  in  T  and  E,  by  the  substitution  of  a  5°  curve. 


PROBLEMS   IN   CHANGE   OF   LOCATION. 


59 


73.  Having  located  a  curve  between  two  tangents  it  is 
required  to  determine  the  radius  of  a  curve  which  from 
the  same  point  of  curve  will  terminate  in  a  defined  parallel 
tangent. 

Let  BE  represent 
the  located  curve  and 
BE'  the  curve  re- 
quired. 

Since  the  tangents 
El  and  ET  are  paral- 
lel the  angles  QBE, 
OEB  and  O'E'B  are 
equal,  the  central  angle 
a  remains  unchanged, 
and  EE'  is  a  prolong- 
ation of  the  chord  BE 


FIG.  29. 


Denote  AE'  the  per- 
pendicular distance  be- 
tween the  tangents  by  d,  and  draw  OD  perpendicular  to  the 
radii  OE  =  R  and  O'E'  =  R'] 

Then 


and 
whence 


(R'  —  R)  cosa  +  R  +  d  =  R', 
R'  (1  —  cosa)  =  R  (1  —  cosa)  -f- 

,_  d 

versine  a 


(32) 


In  practice,  before  removing  the  instrument  from  E,  the  dis- 
tance and  direction  to  E'  should  be  computed  and  a  stake 
driven  there  to  serve  as  a  check  on  the  location  of  the  curve 
BE'. 

The  student  will  observe  that 


EE'  =  d  .  sec 


180  — or 
2       ' 


or, 


EE'  =  d  .  cosec 


(33) 


and  the  angle  which  EE'  makes  with  AE  is 


60  SIMPLE    CUKVES. 

If  BE'  represents  the  located  curve  and  BE  the  curve  re- 
quired, then  R'  will  be  known  and  R  required. 

or,  R  =  R' ? (34) 

versine  a 

If  it  is  determined  on  the  ground  through  what  point  in  the 
prolongation  of  the  chord  BE,  as  E',  the  tangent  must  pass  ; 
then  measure  d'  the  distance  from  E  to  E',  and  proceed  other- 
wise, as  before  — 

E'  =  (fl'  -  P)  cos  a  +  R  +  d'  sin|» 

,,   .    (t 
d  sin— 

or,  R  =  R  + 


I  —  cos  a 


whence       R'  =  E  +  —  -  —  —  (since  1  —  cos  a  =  2  sin2—    •    (35) 
2  sin  i  a  \_  2J 

If  the  parallel  tangent  is  on  the  center  side  of  the  terminal 
tangent. 

(36) 


The  student  -may  find  an  expression  for  Rf  in  terms  of  the 
chords,  BE,  BE'  and  R. 

EXAMPLE.  —  A30  30'  curve  having  a  central  angle  of  34  °  40 
ends  in  a  tangent  IE.  It  is  required  to  substitute  a  curve 
having  the  same  P,C.  but  to  terminate  in  a  parallel  16  feet 
from  IE. 

74.  Having  located  a  curve  between  two  tangents,  it 
is  required  to  determine  the  change  in  the  point  of  curve 
so  that,  with  the  same  radius,  the  curve  may  end  in  a 
given  parallel  tangent. 

The  required  curve  as  to  size  and  shape,  is  the  given  curve  in 
another  position  ;  its  elements  are  precisely  the  same.  Imagin- 
ing the  one  to  merge  into  the  other,  it  will  be  perceived  that  as 
B  Fig.  29a,  approaches  B',  0  will  approach  0',  and  N  and  /  re- 
spectively N'  and  /'.  Hence  the  lines  joining  these  like-lettered 
points,  are  equal  and  parallel.  Therefore  turn  oft'  from  the  ter- 


PROBLEMS    IN    CHANGE    OF    LOCATION.  01 

minal  tangent  at  N,  an  angle  =  «°,  and  measure  in  the  direction 
of  the  telescope  to  a  point  N'  in  the  given  parallel  tangent. 
This  distance  laid  off  from  .Z>  will  give  the  position  of  the 
new  P.C. 

If  the  perpendicular  distance  NK  between  the  parallel  tan- 
gents is  given,  then  in  the  triangle  NKN',  observing  that  the 
angle  NN'K  =  a  and  putting  NK  =  <L 

NN'  =  BE'  =  d  cosec  a.  (37) 

As  in  the  preceding  case  the  direction  of  B'  from  B  is  depen- 
dent upon  the  relative  position  of  the  terminal  and  parallel 
tangents. 


EXAMPLE.  —  A  4°  20'  curve  having  a  central  angle  of  32°  40' 
ends  in  the  tangent  IN.  Compute  the  distance  the  P.C.  must 
be  advanced  along  Bl  so  that  with  the  same  radius  the  curve 
may  be  run  ending  in  a  tangent  parallel  to  IN,  and  16  feet 
farther  from  the  centre. 

BB'  =  16  .  cosec  32°  40'. 
BB'  =  16  X  1.8527  =  29.64. 


62 


SIMPLE    CURVES. 


75.  Having  located  a  curve  between  two  tangents,  it  is 
required  to  change  the  radius  and  the  point  of  curve,  so  that 
the  curve  may  terminate  in  a  given  parallel  tangent  at  a 
point  on  the  same  radial  line  as  the  first. 

Let  /JVbe  the  located  tangent, 
and  I'N'  the  given  parallel 
tangent,  0  and  0"  the  cor- 
responding centers  of  curvature, 
and  R  and  R'  the  radii.  Drop 
a  perpendicular  from  0'  on  OB, 
prolong  the  tangent  BI  and 
radius  ON  to  their  intersecting 
point  P,  and  denote  the  distance 
NN'  on  the  radial  line  by  d. 


Then    PN-PN'  =  d, 

or          (R  —  R')  exsec  a  =  d ; 
d 


FIG.  30. 


whence  (R-R')=. 


exsec  a 


and 


exsec  a 


BB'  =  DO'  =  (R  —  R')  tan  a, 
or  substituting  the  value  of  R  —  R'  from  (38) 


(38) 

(39) 
(40) 


dtan  a 


But 


(41) 


exsec  a 

tang  a  =  exsec  a  cot  \  a, 
.-.  BB'  =  deotgla. 

76.  Having  located  a  curve  between  two  tangents,  it  is 
required  to  find  the  change  in  the  point  of  curve  con- 
sequent upon  a  given  change  in  the  direction  of  the  ter- 
minal tangent  at  the  point  of  intersection,  the  radius 
remaining  the  same. 

Denote  BI  and  B'l  Fig.  31  by  T  and  T'  respectively,  and 
the  angles  NOB  and  N'O'B'  by  a  and  a'  respectively. 

Then  BB'  =  T  -  T'  =  R  (tan  \  a  -  tan i  a').  (42) 


PROBLEMS    IN    CHANGE    OF    LOCATION. 


63 


If  the  position  of  the  P.O.  remains  fixed  and  R  change, 
the  other  condition  as  above  in  this  article,  find  the  new 
radius. 

Calling  a  and  a'  the  angles 
of  intersection,  R  and  R'  radii, 
and  T  the  length  of  the  tan- 
gent. We  shall  have 

T=R  tan i or, 
and         T=R/tan|a/, 
whence 

R'  =  R  tan  $  a  cot  £  a'. 


The  student  may  supply  the 
figure. 


FIG.  31. 


EXAMPLES. 

1.  Given  a  3°  40'  curve,  the  angle  of  intersection  28°  20',  to 
find  the  position  of  the  P.C.,  if  the  terminal  tangent  IN'  makes 
an  angle  of  6°  with  IN;  the  radius  remaining  the  same  length. 

2.  Given  the  same  degree  of  curve,  and  a  as  in  the  last 
example,  to  find  the  change  in  the  radius,  so  that  the  position 
of  the  P.C.  may  remain  unchanged  if  ex.  be  diminished  6°. 

77.  Having  located  a  curve  between  two  tangents,  it  is 
required  to  change  the  radius  and  the  point  of  curve  so 
that  the  terminal  tangent  may  be  changed  in  its  di- 
rection at  the  point  of  tangent. 

Let  IN  represent  the  lo- 
cated tangent  ;  I'N  the  ter- 
minal tangent ;  INF  the  given 
change  in  the  angle  at  N. 
The  student  may  show  that 
a'  =  a  -f  INI'  ;  that 


E'  = 


R  versine  a 
versine  a' 


and  BR  =  Rsma-R'  sin  a'.  (45) 


CHAPTER   IV. 

COMPOUND    CURVES. 
A.    PROBLEMS   IN   LOCATION. 

78.  Given  two  tangents  of  unequal  length,  their  angle 
of  intersection,  and  one  radius,  to  find  the  length  of  the 
radius  of  the  other  branch  of  a  compound  curve  which 
will  unite  the  tangents. 


FIG.  33. 

Let  BPE  represent  the  curve. 

Denote  the  radii  PO  and  PO'  by  B  and  R'  respectively. 
"          tangents  BI  and  IE  by  T  and  T       " 
' «          angles  B OP  and  P O'E  by  a'  and  a"  ' ' 
"          angle  of  intersection  by  a. 


PROBLEMS    IN    LOCATION,  65 

Extend  the  first  branch  of  the  curve  to  D,  where  its  terminal 
tangent  is  parallel  to  the  tangent  IE,  and  draw  DF  parallel  to 
BL  Draw  the  chord  PE,  it  will  pass  through  D.  Draw  also 
01'  and  BD.  In  the  triangle  DEF,  DF  is  equal  to  //'  by  con- 
struction, and,  therefore, 

DF=  T  —  R  t&n$a, 

FE=  T'  -  Btanio-, 
and  angle  F  =  180°  -  a, 

whence  the  angle  FED  and  side  ED  may  be  found.     Now 
2  FED  —  a",  and  hence  a'  =  a  —  a"  becomes  known. 

Again  2R  sinFED  =  PD, 

2R'sinFED  =  PE, 
Therefore  2  R  '  sin  FED  =  2  R  sin  FED  +  DE, 

whence  ir  =  B  +  __.  (40) 


The  point  D  may  be  located  by  turning  off  half  a°  from  the 
P.C.  on  tangent  BI,  and  measuring  the  corresponding  long 
chord.  Then,  having  computed  the  angle  FED,  set  up  the 
instrument  at  D,  turn  off  from  the  parallel  tangent  through 
D  the  angle  FED,  and  thereby  locate  P,  the  P.C.C.,  and  E,  the 
P.T.,  after  which  in  the  usual  manner  the  curve  may  be  staked 
out. 

EXAMPLES. 

1.  Given  BI  and  El  500  and  600  feet  respectively,  their 
angle  of  intersection  21°  40',  and  the  radius  of  the  first  branch 
2500  feet.     Find  the  radius  of  the  second  branch,  and  number 
the  P.C.C.  and  P.T.,  assuming  the  P.C.  to  be  24  -f  60. 

2.  The  student  may  reason  out  by  aid  off  a  diagram  the 
case  where  the  length  BI  is  greater  than  El,  and  the  given 
radius  of  the  first  branch  is  the  longer  ;  and  verify  his  deduc- 
tions by  numerical  example  and  construction. 

79.  Given  the  length  of  the  straight  line  between  the  P.C. 
and  the  P.T.,  the  angles  which  it  makes  with  the  tangents, 
and  the  radius  of  the  first  branch,  to  find  the  radius  of  the 
second  branch  of  a  compound  curve  uniting  the  tangents. 

In  Fig.  33  let  d  denote  the  distance  from  B  to  E,  B  and  E 
the  angles,  at  the  respective  points,  between  the  chord  and  the 


66 


COMPOUND   CURVES. 


tangents,  R  and  R'  the  radii  of  the  first  and  second  branches 
respectively,  and  a'  and  a"  as  shown  in  the  figure.  The  angle 
IBD  =  $a  =  ±(B  +  E).  In  the  triangle  BDE,  BE  is  given,  ED 
=  2R  sin  i  (B  +  E)  and  the  angle  DEE  =  E  —  \(E  +  E}=± 
(B  —  E).  Therefore  the  angle  BED  and  side  DE  can  be 
computed,  and  E — BED=  the  angle  which  the  chord  PDE 
makes  with  the  tangent  IE,  and  thus  a."  and  of  become  known. 
Then,  as  before,  using  the  relation  between  an  angle,  its  sub- 
tended chord,  and  radius,  the  value  may  be  found  of 


DE 


2  sin  PEI 


(47) 


which  agrees  with  (46). 


80.  Given  the  radii  and  the  central  angles  of  a  compound 
curve  uniting  two  tangents,  to  find  the  lengths  of  the  tan- 
gents, the  line  connecting  the  P.O.  and  P.T.,  and  the  angles 
which  this  line  makes  with  the  tangents. 

In  Fig.  34  let 
B  and  E  represent 
the  P.C.  and  P.T. 
respectively,  P  the 
P.C.C.jOandO'the 
centers,  IB  and  IE 
tangents,  d  the  con- 
necting line,  and 
other  notation  as 
shown  in  the  fig- 
ure. With  the 
given  central  an- 
gles and  radii  com- 
pute the  lengths 
of  the  chords  BP 
and  PE  ;  find  also 
the  angles  OPE  = 
OBP  and  0'PE  =  0'EP.  Then  in  the  triangle  BEP  find 
the  angles  PEE  and  PEE  and  the  side  BE  =  d.  The  un- 
known angles  may  now  be  found  as  follows  :  — 


PROBLEMS   IN   LOCATION. 


6T 


The  angle  EBI  =  9Q  —  PBO  +  PEE. 
The  angle  BE  I  =  9Q  —  PEO'  +  PEE. 

Then  in  the  triangle  BE  I  having  all  the  angles  and  the  side 
d,  compute  the  tangents  El  and  IE. 

EXAMPLE.  —  Given  R,  1432.7  feet,  R',  2148.8  feet  a',  20° 
and  a",  45°  ;  to  find  the  lengths  of  the  tangents,  the  line  BE, 
and  the  angles  which  the  tangents  make  with  BE. 

81.  Given  the  length  of  the  straight  line  between  the 
P.O.  and  the  P.T.,  and  the  angles  which  it  makes  with  the 
tangents,  it  is  required  to  find  the  radii  of  a  compound 
curve  having  the  common  tangent  parallel  to  the  straight 
line. 

In  the  figure  let 
DF  represent  the 
tangent  parallel  to 
the  given  line  BE. 
PtheP.C.C.,  ZXBE 
and  FEE  the  given 
angles,  and  d  the 
length  of  BE. 

It  is  evident  that 


=  FP  and  the  an- 
gles DBF  and  FEP 
=  %  B  and  £  E  re- 
spectively. 

In  the  triangle 
BEP  we  have  the 
proportion 


BP:  BE  =  sin  BEP:  sin  [180  -  \  (B  +  E)], 
dsinlE 


E) 


and 


PE  =  - 


E) 


68 


COMPOUND   CURVES. 


Having  found  the  chords  of  the  branches  the  radii  may  be 
calculated  by  the  well  known  relation  between  the  radius, 
central  angle,  and  chord,  which  gives 


sin  i  B      sin  \ B  .  sin  £  (13  +  E)' 


and 


(48) 


(49) 


sin i#.  sin 

EXAMPLE.  —  Given  the  distance  between  the  P.C.  and  P.T. 
1000  feet,  angle  B  12°  30',  E  14°  40'.  Find  the  radii  R 
and  R'. 

B.    OBSTACLES. 

82.  To  locate  a  compound  curve  when  the  P.C.C.  is  inac- 
cessible ;  the  central  angles  of  and  a",  and  the  degree  of  each 
branch  of  the  curve  being  given. 


If  the  P.C.  and  P.T.  are  determined,  the  curve  may  be  staked 
out  from  these  points. 

Or,  if  the  obstructions  do  not  extend  far  from  the  P.C.C.,  as 
indicated  in  Fig.  36,  set  as  many  stakes  on  the  first  branch  as 


OBSTACLES.  69 

possible  ;  calculate  the  chord  B^m,  parallel  to  the  common 
tangent  through  P,  and  the  length  of  the  middle  ordinate  mP 
for  double  the  arc  B^P.  Then  in  the  second  branch  find 
the  angle  ft"  having  the  same  middle  ordinate  mP,  for 
double  its  arc,  whence  the  length  of  the  chord  mE,  may  be 
computed. 

Then  deducting  a!  -f  ft'  from  a  the  angle  E^O'E  will  be 
obtained,  and  thence  the  distance  to  the  P.T. 

If  the  curves  are  much  obstructed,  so  that  few,  or  no  stakes, 
can  be  set  in  them,  run  direct  from  the  P.O.  to  some  point  on 
the  second  branch  —  obtaining  the  necessary  data,  as  indicated 
in  the  foregoing.  Or,  if  it  is  more  practicable,  run  the  com- 
mon tangent  APD. 

The  length  of  this  line  can  readily  be  found,  since  AP  =  AB 
and  PD  =  DE  are  tangent  distances,  by  applying  the  well- 
known  rule  :  —  Radius  multiplied  by  tangent  of  one-half  the 
central  angle  is  equal  to  the  tangent  distance. 

EXAMPLES. 

1.  Given  a'  =  12°,  a"  =  10°  30',  and  the  degrees  respectively 
3  and  5  to  locate  the  P.T.,  the  P.C.C.  being  obstructed  by  a 
building  : 

BA  =  AP  =  1910  tan  6°  =  201.6  feet, 
PD  =  DE  =  1146  tan  5°  15'  =  105.3  feet. 

With  the  instrument  at  B,  set  as  many  stakes  as  possible  on 
the  curve  BP,  then  sight  along  the  tangent  and  measure  BA 
201.6  feet  ;  remove  to  A,  turn  off  the  angle  a',  and  measure 
AD  306.9  feet,  offsetting  at  P.  With  the  instrument  at  1), 
turn  off  the  angle  ot"  as  indicated  in  the  figure,  and  measure 
DE  105.3  feet  to  the  P.T.  Having  now  located  the  P.T.,  if 
practicable,  set  one  or  two  stakes  in  the  curve  EP. 

The  student  may  determine  where  to  place  these  last 
stakes  and  how  to  number  them,  that  they  may  appear 
consecutive,  the  P.O.  being  at  number  42.  State  also  the  plus 
of  the  P.T. 

2.  Given  a  2°  and  a  4°  curve,  of  and  a"  respectively  9°  and 
13°,  conditions  favorable  to  setting  the  first  three  stakes  on 


70 


COMPOUND    CURVES. 


first  branch,  and  from  the  third  station  run  a  parallel  to  the 
common  tangent  past  the  obstacle.  Show  how  to  continue 
the  work,  find  the  point  where  the  parallel  line  intersects  the 
second  branch,  the  reading  of  the  instrument  at  this  point 
when  telescope  sights  the  tangent,  supposing  the  index  was  at 
zero  at  the  P.C.,  and  find  the  plus  of  the  P.T.,  calling  the  P.C. 
17  +  40. 

C.     PROBLEMS   IN   CHANGE    OF   LOCATION. 

83.  Having  located  a  compound  curve  terminating  in  a 
tangent,  it  is  required  to  change  the  location,  so  that,  with 
the  same  radii,  the  curve  may  end  in  a  parallel  tangent  at 
a  given  perpendicular  distance  from  the  terminal  tangent. 

FIRST  CASE.  —  The  parallel  tangent  farther  out  than  the 
terminal  tangent,  and  the  second  branch  of  the  curve  having 
the  longer  radius. 

The  figure  opposite 
represents  the  case. 
The  first  curve  with 
center  0  having  been 
run  from  A,  com- 
pounded at  P  into 
one  with  radius  O'P, 
and  terminating  in  a 
tangent  at  T.  It  is 
desired  to  locate  the 
tangent  further  out 
at  a  given  perpendic- 
ular distance  from  the 
present  one.  It  is 

evident  that  the  P.C.C.  must  be  moved  back  on  the  sharper 
curve  to  some  point  P'.  The  practical  question  is  to  find  this 
point. 

Let  0"P'  represent  the  required  position  of  the  longer 
radius,  and  draw  OMN  perpendicular  to  the  radii  O'T  and 
O"T.  It  is  evident  that  the  angle  POP'  is  equal  to  a'  —  a  = 
OO"N —  OO'M,  and  as  the  latter  is  known  we  may  find  the 
former  as  follows  :  — 


PROBLEMS    IN    CHANGE    OF    LOCATION.  71 

Let  R  denote  the  radius  of  the  first  branch  of  the  curve. 
"    R'     "         "         "         "       second     "         "         " 
"    a      "         "    angle       "       OO'M. 
»    a'     "         "         "         "       00"N. 

"    d      "         "    perpendicular  distance  between  the  tangents. 
Then  (R'  -  R)  cos  a  =  O'M, 

and  ( R'  -  R)  cos  a'  =  0"N. 

But  O'M —  O"N  =  d  substituting  we  obtain 

(R'  —  R)  cos  a  —  (R'  —  R)  cos  a'  =  d, 

or,  cos  a.'  =  cos  a  — (50) 

R'  —  R 

and  a'  —  a  =  POP'. 

Divide  a'  —  a  by  the  degree  of  the  first  branch  and  thus 
ascertain  the  distance  from  P  to  P'. 

To  find  the  distance  and  direction  of  T'  from  T;  connect 
the  points  O'O",  and  draw  a  perpendicular  from  O  to  the  line 
O'O"]  also  draw  the  perpendicular  O'K  on  the  radius  0"T' 
prolonged.  Then,  evidently, 

TT  =  O'O"  =  2(R'-  R)  sin^a'  -  a).  (51) 

Now  considering  the  angles  about  the  point  0',  we  perceive 
that 

90  +  a  =  KO'O"  +  90-|  (a'  -  a), 
or,  KO'  O"  =  a  +  \  (a'  -  a) ; 

.-.  angle  STT  =  KO'O"  =  ^(a'  +  a).  (52) 

With  the  instrument  then  at  T",  turn  off  from  the  tangent 
an  angle  equal  to  the  arithmetical  mean  of  the  given  and  com- 
puted angles;  the  telescope  will  then  point  in  the  direction  of 
the  new  P.T.  Measure  off  the  distance  computed  by  (51) ;  set, 
and  center  a  stake  there  before  running  in  the  new  curve. 

REMARK.  —  This  method,  based  on  the  arithmetical  mean, 
for  obtaining  direction  between  the  tangent  points,  holds  true 
for  all  four  of  the  cases  coming  under  this  head.  It  was  sug- 
gested by  Mr.  Edward  Godfrey,  Class  of  '93,  W.U.P. 

EXAMPLE.  —  Given  a  5°  curve  compounding  into  a  2°  40'  for 
12°  50',  and  terminating  in  a  tangent;  it  is  desired  to  move  the 
tangent  out  10  feet.  Ascertain  the  change  necessary  in  the 
P.C.C.  Ans.  47  feet  towards  P.O. 


72 


COMPOUND   CURVES. 


SECOND  CASE.  —  The  parallel  tangent  farther  in  than  the 
terminal  tangent  and  the  second  branch  of  the  curve  having 
the  longer  radius. 


O' 


FIG.  38. 


Adopting  the  notation  of  the  first  case  (R'  =  radius  of  second 
branch)  we  have, 

(R'  —  R)  cos  of  —  (R'  —  R)  cos  a  =  d  ; 
d 


whence 


and 


cos  a  =  cos  a 


IF-JB' 


(53) 


Divide  as  before  this  angle  by  the  degree  of  the  first  branch, 
and  thereby  ascertain  the  distance  to  P' ',  or  how  much  the 
sharper  branch  must  be  lengthened. 

Again  connect  CfO"  and  the  equidistant  points  TT'  and 
draw  perpendiculars  from  0'  to  0"T'  and  from  O  to  O'O"  and 
as  before  we  shall  have 

TT  =  (y  0"  =  2  (R  -  R)  sin  \  (a  -  a') ,  (54) 

and  K&O"  =  i  (a  +  a'}.  (55) 

EXAMPLE.  —  Given  a  5°  curve  compounding  into  a  3°  for 
12°  and  terminating  in  a  tangent.  It  is  desired  to  move  the 
tangent  in  8  feet.  Show  that  the  new  P.C.C.  is  67  feet  farther 
from  the  P.C.,  and  locate  the  required  P.T, 


PROBLEMS   IN   CHANGE   OF   LOCATION. 


73 


THIRD  CASE.  —  The  parallel  tangent  farther  out  than 
the  terminal  tangent,  and  the  second  branch  of  the 
curve  having  the  shorter  radius. 

It  is  evident  that  the  flat- 
ter branch  must  be  length- 
ened at  the  expense  of  the 
sharper  one,  and,  as  before, 
the  angle  POP'  must  be  de- 
termined in  order  to  locate 
Pf. 

Denote   the   angle   O'OM 
by  a,   0"ON  by  «',  and  R 
and  7t',  the  radii  of  the  first 
and  second  branches  respec-    0 
tively. 


Then 


or, 
and 


FIG.  39. 


(R  —  R')  cos  a'  =  (R-  R')  cos  a  +  d, 
d 


cos  of  =  cos  a 


a  —  a'  =  POP'. 


R 


(56) 


Whence  the  distance  PP'  may  be  found  as  before,  and  the 
first  branch  extended. 

The  distance  between  the  tangent  points  may  be  determined 
as  in  the  second  case  thus, 


TT'  =  O'O"  =  2R  —  R' 


a  —  a'). 


(57) 


The  student  may  show  that  the  direction  of  T'  from  T  is 
obtained,  as  in  the  other  cases,  by  deflecting  from  the  tangent 
at  !T,  Ka  +  «> 

EXAMPLE.  —  Given  a  3°  20'  curve  compounding  into  a  5°  40' 
for  22°,  and  terminating  in  a  tangent.  It  is  desired  to 
have  the  tangent  16  feet  farther  out.  Locate  the  new 
P.C.C.,  and  the  new  P.T.,  and  state  the  length  of  the  curve 
required. 


74 


COMPOUND    CURVES. 


FOURTH  CASE. — The  parallel  tangent  farther  in  than  the 
terminal,  and  the  second  branch  of  the  curve  having  the 
shorter  radius. 


FIG.  40. 

Adopting   the  notation  of   the  third  case  (R 
second  branch)  we  have 

(R  —  R')  cos  a^  =  (R  —  R')  cos  a  —  d, 
d 


or, 
and 


cos  a'  =  cos  a  — 


R-  R'' 


=  radius  of 


(58) 


Whence  the  distance  from  P  to  P',  the  new  P.C.C.,  is  readily 
obtained. 

The  distance  between  the  tangent  points  is  shown  by  the 
equation 

TT'  =  (yO"  =  2(R-  R')  sin  £  (a'  -  a).  (59) 

and   the    angle    to    be  turned  off   from  the  tangent  at   T  = 
i  (X  +  "> 

EXAMPLE.  —  Given  a  2°  curve  compounding  into  a  3°  for 
10°,  and  terminating  in  a  tangent.  It  is  desired  to  move  the 
tangent  in  12  feet.  Show  that  the  P.C.C.  has  to  be  moved 
176.5  feet,  and  give  the  direction  and  distance  of  the  new  P.T. 
from  T. 


PROBLEMS    IN    CHANGE    OF    LOCATION. 


75 


84.  A  method  of  solution,  similar  to  the  preceding,  may 
be  applied  to  the  following  problem: 

Given  a  simple  curve  AT  terminating  in  a  tangent  at 
T\  it  is  required  to  find  the  point  P,  at  which,  by 
compounding  with  a  curve  of  known  radius,  the  curve 
may  end  in  a  given  parallel  tangent. 

A 


Then 
or, 

whence 


FIG.  41. 

Denote  the  unknown  angle  POT  by  a, 

"          perp.  dist.  between  tangs,  by  d, 

"         first  radius  OTby  J?, 

"         second  radius  (XT'  by  R'. 

00/cosnr=  O'JW, 
(fi'  -  R)  cosa  =  R'  -(R  +  d); 
d 


cos  a  =  1  — 


R'  —  R 


(60) 


EXAMPLES. 

1.  Given  a  5°  20'  curve  terminating  in  a  tangent  at  T\  it  is 
required  to  locate  a  point  P,  whence  a  3°  30'  curve  may  be 
run  which  shall  end  in    a   parallel  tangent  18'  farther   out. 
Ans.  a  =  14°  32',  and  distance  from  T  to  the  P.C.C.  =  272.5  ft. 

Solve  the  following,  making  such  modifications  of  the  pre- 
ceding formula  as  may  be  necessary. 

2.  Given    a  2°  40'    curve   terminating   in    a  tangent;    it  is 
required  to  locate  a  point  P,  whence  a  4°  20'  curve  may  be  run 
which  shall  end  in  a  parallel  tangent  16'  farther  in. 


76 


COMPOUND   CURVES. 


85.  Having  located  a  compound  curve  terminating  in 
a  tangent,  it  is  required  to  change  the  P.C.C.  and  the 
radius  of  the  second  branch,  so  that  the  curve  may  end 
in  a  parallel  tangent  at  a  given  point  on  the  same  radial 
line. 

A 


FIG.  42 


FIRST  CASE.  —  The  parallel  tangent  outside  the  terminal 
tangent,  and  the  second  branch  of  the  curve  having  the 
longer  radius. 

Let  APT  be  the  located  curve,  and  AP'T'  the  required 
curve,  having  the  centers  0  and  0'  respectively,  and  P'  the 
new  point  of  compound  curvature. 

Let  7?  =  OP,  the  radius  of  the  nrst  branch, 
«  R'  =  O'P,         "  "         second      " 

"  R"  =  0"P'        "  "         required  " 

"     <1  =  the  given  distance  between  tangents, 
and  the  angles  as  indicated  in  the  figure. 

Extend  the  first  branch  of  the  curve  from  P  till  it  termi- 
nates at  E  in  a  tangent  parallel  to  the  terminal  tangent  TF. 
Draw  the  radius  OE,  and  with  the  radius  0"P'  describe  the 
arc  P'T.  The  angle  POE  being  equal  to  PO'T,  the  chord 
PT  will  pass  through  E  ;  and  the  angle  P'OE  being  equal  to 


PROBLEMS   IN   CHANGE   OF   LOCATION.  77 

P'O"T',  the  chord  P'T  will  also  pass  through  E.  The  point 
Pf  may,  therefore,  be  constructed  by  extending  the  chord  from 
T'  through  E  till  it  meets  the  first  branch. 

There  are  two  principal  steps  in  the  solution. 

First  to  find  a?,  and  with  it,    second,  to  find  R". 

To  find  a'  the  figure  shows  that  the  angle  P'T'K  is  equal  to 
£  a',  hence, 


But  —  =  tan  i-  a  and  T'K  --=  TF=OM=  (R'  —  R)  sin  a. 

TF 

Substituting,  there  results, 

(61) 


In  the  triangle  00'  0",  by  the  law  of   sines   00"  may  be 
found,  which  added  to  OP',  will  give  the  length  of  R". 
Or,  in  the  triangles  OO"M  and  OO'M  find  at  once 

(R"  -  R)  sin  a'  =  (R'  -  R)  sin  a; 


whence  E"  =  (R'  -  R)         >  +  A  (62) 


SECOND  CASE.  —  The  parallel  tangent  inside  of  the  terminal 
tangent  and  the  second  branch  having  the  longer  radius. 

Using  the  same  figure  and  notation  as  in  the  preceding,  a  and 
R'  become  the  unknown  quantities,  and  we  have 

tanl^™-* 
T'K    ' 

or,  tan  \  a  =  tan  |  a'  —         _        . ; »  (63) 


sin  a.' 
and  B'^K"  _*)__+£.  (64) 


EXAMPLES. 

1.  Given  a  5°  20'  curve  compounded  at  P,  Fig.  42,  into  a 
2°  40'  and  terminating  in  a  tangent  FT  at  71,  making  a:  16°  30'; 
it  is  required  to  end  in  a  parallel  tangent  KT'  intersecting  the 
prolongation  of  O'T  18  feet  farther  out.  Find  P',  and  O"P', 
the  new  P.C.C.  and  radius  respectively. 


78 


COMPOUND    CURVES. 


2.  Given  a  5°  40'  curve  compounded  into  a  2°  30',  and  ter- 
minating in  a  tangent  making  a'  32°  40';  it  is  required  to  end 
the  curve  in  a  parallel  tangent  intersecting  the  same  radius 
12  feet  farther  in.  Locate  the  new  P.C.C.,  and  determine  the 
radius  of  the  last  branch. 

THIRD  CASE.  —  The  parallel  tangent  outside  the  termi- 
nal tangent  and  the  second  branch  of  the  curve  having 
the  shorter  radius. 


FIG  43. 

Let  R  =  OP,  the  radius  of  the  first  branch, 
«  R'  —  O'P,         "  "         second     " 

«  R"  —  0"P',       "  "         required " 

«     d  =  distance  between  tangents,  and  the  angles  as 
shown  in  the  figure,     a'  and  R"  are  to  be  determined. 
=  EF  =  EK  —  d 

-  rF~     TK   ' 


and 


whence 


(R  —  R")  sin  a'  =  (R  — 


R"  =  R  —  (R  —  R'} 


sin  a: 


(65) 


(66) 


PROBLEMS   IN   CHANGE   OF   LOCATION.  79 

FOURTH  CASE.  —  The  parallel  tangent  inside  the  termi- 
nal tangent  and  the  second  branch  having  the  shorter 
radius. 

Using  the  same  notation  as  in  the  preceding  case,  a  and  R 
become  the  unknown  quantities,  and  we  have 


or,       g  tan^  =  tani«'  +  (fi_fi)s.na/,  (67) 

and          (R  —  R")  sin  a'  =  (R  —  R')  sin  a  • 

sin  ccf 
whence  R'  =  R-(R-  R")  -         •  (68) 


EXAMPLES. 

1.  Given  a  2°  50'  curve,  compounded  at  P,  Fig.  43,  into  a 
4°  50',  and  terminating  in  a  tangent  at  T,  making  a  30°  ;  it  is 
required  to  end  the  curve  in  a  parallel  tangent  intersecting 
the  prolongation  of  O'T  at  T',  24  feet  distant  from  T.     Find 
P'  and  the  length  of  0"P'. 

2.  Given  a  2°  40'  curve  compounded  into  a  4°  40',  and  ter- 
minating in  a  tangent,  making  a'  24°  ;  it  is  required  to  end  in 
a  parallel  tangent,  intersecting  the  same  radius  20  feet  farther 
in.     Locate  the  new  P.C.C.,  and  determine  the  radius  of  the 
last  branch. 

86.  Having  located  a  compound  curve  between  two 
tangents,  it  is  required  to  shift  the  P.C.C.  and  change  the 
radius  of  the  last  branch  so  that  the  curve  may  end  at  some 
other  point  in  the  terminal  tangent. 

FIRST  CASE. — When  the  second  branch  of  the  curve  has 
the  longer  radius  and  the  point  in  the  tangent  is  given 
as  at  E'. 

In  the  figure  there  are  given  the  central  angles  at  0,  and  O', 
the  radii  drawn  from  these  points,  and  the  distance  d  between 
E  and  E'.  It  is  required  to  find  the  angle  at  O"  and  the 
radius  0"E'. 


80 


COMPOUND   CURVES. 


Extend  the  first  branch  of  the  curve  to  D,  where  its  terminal 
tangent  becomes  parallel  to  IE  ;  draw  the  chord  PE  ;  it  will 
pass  through  D,  and  the  line  drawn  from  E'  through  D 
prolonged  to  the  curve  will  indicate  at  P'  the  new  P.C.C. 
(See  Art.  73.)  Finally  draw  from  0,  the  line  OMN  perpen- 
dicular to  R'  and  R",  and  with  center  0'  describe  the  arc  OQ. 


By  construction  DF=  MQ. 

.-.  DF=  (R'  -R)(l-  cos  a') 
=  (R'  —  R)  versine  a'. 

In  the  triangles  DEF  and  DE'F, 


<=§- 


whence,  by  substitution, 


2 


or, 


, 

DF1 


cot—  =  cot—  + 

9  9.      '     I  ft"  — 


(70) 


(R'  —  R)  versine  a' 
Divide  a'  —  a''  by  the  degree  of  curve  of  first  branch,  and 
the  result  will  show  how  far  to  extend  the  first  branch  to 
reach  the  new  P.C.C. 


PROBLEMS   IN   CHANGE   OF   LOCATION.  81 

In  the  triangles  OO'M  and  00"N, 

(R/  —  R)  sin  a'  =  OM , 
(R"-  R)sina"=ON, 
or,  by  subtraction, 

(R"  —  R)  sin  a?'  —  (R  —  R)  sin  a'  =  MN.  =  d ; 

(R'  —  R)  sin  a'  +  d  ,--,  v 

whence,  JJ"  =  B  +  L         si'n^  <71> 

Assuming  the  terminus  of  the  curve  at  E',  write  the  equa- 
tions for  a'  and  R'  so  that  the  P.T.  may  be  at  E,  d  distance 
from  E'. 

SECOND  CASE.  —  Conditions  the  same  as  in  the  preceding, 
except  the  point  is  not  definitely  located ;  it  must  be,  how- 
ever, somewhere  on  the  terminal  tangent. 

Here  we  may  assume  new  central  angles,  that  is,  a  new 
P.C.C.,  and  calculate  Jf2";  or,  we  may  assume  R" ',  and  compute 
the  change  in  the  angles.  The  student  may  show  that,  assuming 
the  P.C.C., 

R"  =  R+ ?^,  (72) 

versme  a'' 

and,  assuming  R", 

veisinea"  =  -^-.  (73) 

xt    —  _tt 

Show  also  how  to  find  the  distance  d  in  each  of  the  exam- 
ples of  this  case,  so  that  in  practice  a  check  may  be  had  on 
the  work. 

EXAMPLE.  —  Having  located  a  compound  curve  terminating 
in  a  tangent,  the  radii  of  the  first  and  second  branches  respec- 
tively, 1600  and  2500  feet,  and  the  angle  a  =  32°;  it  is  required 
to  move  the  P.C.C.  back  150  feet.  Find  the  radii  of  a  curve 
which  shall  end  somewhere  in  the  terminal  tangent,  and  locate 
the  new  P.T. 

87.  To  substitute  a  three-centred  compound  curve  for  a 
simple  curve. 

In  the  figure  let  a  denote  the  central  angle  at  0,  a'  the 
angle  at  0',  R,  R',  R",  R",  the  radii,  the  last  two  being  equal, 
and  sweeping  equal  arcs  P'E  and  BP.  It  is  evident  that 


82 


COMPOUND    CURVES. 


the  intersection  0'  of  the  radii  n"R"  will  be  on  the 
line  bisecting  the  simple  curve,  and  (7  is,  therefore,  the  cen- 
ter of  the  middle  part  of  the 
compound  curve,  0"O"'  being 
centers  of  the  other  parts, 
and  a  =  the  sum  of  the  angles 
at  the  centres  O/  0"  and  0'". 
In  practice  we  assume  the 
radii  R'  and  R",  or,  R'  and 
the  equal  angles  at  0"  and 
0"' ',  and  compute  what  is  re- 
quired. 

Proceeding  under  the  first 
assumption,  and  using  the 
triangle  0,  0',  0"  (0,  0',  0'" 
would  answer  as  well),  we 
have  the  proportion, 


R"  —  R  :  R"  —  R  =  sin^ :  sin^- ; 


whence, 

and  the  angle  at 


O"  =  0'"  = 


(74) 


(75) 


Under  the  second  assumption  we  find  from  the  above  pro- 
portion, 

(R"  -  R)  sin^  =  (R"  —  R'}  sin^ ; 


whence, 


JR"  = 


sin     - sin 

2  2 


(76) 


a:'  being  known,  since  all  the  other  angles  are  given. 
EXAMPLE. —  Given,     R  =1910,      a  =  36°, 

#'  =  1508,    £"  =  5730. 
Find  EP'  =  PB.  Ans.    177ft. 


CHAPTER   V. 


MISCELLANEOUS    PROBLEMS. 

88.  Given  a  simple  curve  intersected  by  a  straight  line ; 
it  is  required  to  find  a  point  on  the  curve  from  which  to  run 
a  curve  of  given  radius  that  shall  terminate  in  the  straight 
line  as  a  tangent. 

FIRST  CASE.  —  The  P.T.  on  the  straight  line  inside  of  the 
given  curve.  Let  BE  represent  the  curve,  TN  the  straight 
line,  and  N  the  point  of 
intersection.  Measure  for- 
ward and  backward  from 
N  to  points  equidistant  on 
the  curve;  bisect  the  line 
connecting  these  points, 
and  thus  obtain  the  direc- 
tion of  the  radius  ON. 
Or,  more  accurately,  with 
the  instrument  at  N  and 
the  reading  zero,  direct 
the  telescope  to  a  point  in 
the  curve  100  feet  distant,  and  turn  off  from  the  curve  an 
angle  equal  to  one-half  the  degree  of  the  curve,  then  the 
telescope  will  point  in  the  direction  of  the  tangent  NG. 
Observe  the  angle  ONT.  Suppose  the  curve  produced  to  E' 
where  its  terminal  tangent  becomes  parallel  to  the  given  line 
NTj  draw  O'M  perpendicular  to  OE'.  Denote  the  radii  of  the 
given  and  required  curves  by  R  and  R'  respectively,  the  known 
angle  NO  Thy  a,  and  the  required  angle  PO'P'  by  p.  P  being 
the  point  sought. 

It  is  evident  from  the  figure  that, 

R  cos  a  =  (R  —  R)  cos  /?  +  JB' ; 

Rcosa—  R' 
or  cos/3=        R  —  K '  (77) 

and  yg  —  a  =  angle  PON, 

whence  P  may  be  located,  and  the  curve  PD  set  out. 


84  MISCELLANEOUS    PROBLEMS. 

SECOND  CASE. — The  P.T.  on  the  straight  line  outside  of 
the  given  curve. 
D 


In  Fig.  47,  a  is  found  as  before,  and  ft  is  to  be  determined. 
Similarly  as  in  the  preceding  case,  we  have 


and  (R  +  R')  cos  ft  +  R  =  OT, 

or,  R  cos  a  =  (R  +  R')  cos  (3  +  R', 

Rcosa  —  R' 


whence 


, 


and  ft  —  OL  =  the  angle  PON,  with  which  P  may  be  located, 
and  the  curve  PD  set  out. 

89.  Given  a  tangent  TT',  and  a  curve  TV,  it  is  required  to 
connect  these  by  a  curve  VT'  of  known  radius  forming  a  Y 
at  some  point  V.  The  tangent  points  T',  V,  and  the  angles 
a  and  a'  are  to  be  determined. 

In  Fig.  48,  suppose  P  V  drawn  from  the  middle  of  TT'  tan- 
gent to  the  curves  at  V.  This  construction  makes  P  T,  P  V,  and 
PT'  equal.  Connect  the  centers  0  and  0'.  Now  since  the  sup- 
plementary angles  TP  V  and  T'P  V  are  bisected  respectively  by 
OP  and  O'P  the  angle  OPOf  is  a  right  angle  and  PV  is  there- 
fore a  mean  proportional  between  the  radii  R  and  R'. 

Or  denoting  PV  by  x  and  the  angles  at  0  and  0'  by  a 
and  of 


TO   LOCATE   A   Y. 


85 


hence 
and 


tania  =  -=A/^ 


(79) 
(80) 


a7 


180  -  a. 
P 


\       - 


FIG.  48. 

90.  Given  a  curve  TT  located,  and  the  radii  of  two 
other  curves,  it  is  required  to  connect  the  system  forming  a 
Y,  as  indicated  in  figures  49  and  50. 


FIG.  49. 

FIRST  CASE. — The  curves  being  convex  to  each  other,  as 
in  figure  49.     Connect  the  centers  of  the  curves  O,0',O",  thus 


86 


MISCELLANEOUS   PROBLEMS. 


forming  a  triangle  in  which  the  three  sides  are  known,  and 
compute  the  angles  and  thence  the  common  tangent  distance. 
Practically  find  one  of  the  central  angles  as  <*,  then 

Tangent  distance  =  x  =  R  tan  i  a  =  PT=PT=PV 


tan 


n'—x- 
a-R'~ 


and 


tan  i  « 


(81) 


(82) 


R 


whence  the  limiting  points  of  the  curves  are  determined. 

EXAMPLE.  — Given  TT',  a  3°  20'  curve,  and  TV,  a  4°  40' 
curve,  it  is  required  to  connect  them  by  VT'  a  6°  curve.  Com- 
pute the  central  angle  and  the  common  tangent  distance. 

SECOND  CASE.  —  The  curves  being  convex  in  the  same 
general  direction.  Fig.  50. 

OT=R 

0'T=Rf 

O"T  =  R" 

Connect  the  centers 
O  O'  O"  of  the  curves 
thus  forming  a  tri- 
angle and  compute  the 
central  angles  and 
common  tangent  dis- 
tance as  in  the  first 
case. 

The  student  may 
show,  by  construction, 
how  to  locate  P,  the 
intersecting  point  of 
the  tangents. 

EXAMPLE.  —  Given 
TT  a  1°  20'  curve, 
and  TV  a  7°  curve,  to 

connect  by  an  8°  30'  curve  VT'.     Compute  the  central  angle 

and  the  common  tangent  distance. 


A    TRACK   WITH    CIRCULAR    ENDS. 


87 


91.  To  lay  out  a  track  of  a  given  length,  having  cir- 
cular ends  connected  by  two  tangents  of  known  direc- 
tions, and  of  given  distance  apart  at  either  end. 


FIG.  51. 

In  the  figure  denote  the  line  AB  by  &,  the  known  angles 
FAG  and  AFG,  as  found  from  the  direction  of  the  tangents, 
by  a  and  ft  respectively.  Call  DB,  y;  DP,  x  ;  DC,  c;  BF,  R ; 
and  DE,  r. 

Then  the  distance  round,  or  length  of  track, 


(83> 


Now, 


E  =  "  sec  a, 


r  =  -  sec  a, 


x  =  -  cosec  a, 
y  +  x  =  -  cosec  a, 


V=(-.  —  -)  cosec  a. 
Substituting  in  (83),  we  obtain 


-    cosec  a 


(84) 


in  which  c  is  the  only  unknown  quantity. 


MISCELLANEOUS   PROBLEMS. 

EXAMPLE.  —  If  AB  =  50Q,  the  distance  round  £  mile  and 
ft  =  80°. 
Then,        2640  =  2  [-  TT  .  250  sec  10°  +  f  250  -  -\  cosec  10° 


From  which  c  may  be  found,  thence  the  arcs  DMC,  ANB,  and 
the  tangent  DB. 

92.  Given  two  curves  united  by  a  tangent,  to  substitute 
for  the  tangent  a  simple  curve  of  known  radius,  com- 
pounded with  the  others. 


Denote  by  R  the  radius  OT, 

«  R'  "  O"T, 

R"  «  0"P  =  0"P', 

"  d  the  distance  between  tangent  points  TT', 

«  a  the  angle  TOP, 

«  £  "  DOO'. 

Draw  O'D  perpendicular  to  OT. 


Then 


and  OO'  =  (R-R')secj3. 

Now  in  the  triangle  00'  0",  the  sides  being  known,  find  the 
angle  0,  hence  the  angle  <*,  and  thereby  locate  the  point  P. 
In  a  similar  manner  the  point  P'  may  be  located. 


TANGENT   TO   CURVE   FROM   POINT. 


89 


Can  a  radius  as  short  as  i  (R  -f-  R'  -f  0(7)  be  employed? 
If  R  =  R',  show  that 


EXAMPLE.  —  Given  two  curves,  TP  and  T'P',  of  4°  40'  and 
5°  50'  respectively,  connected  by  a  tangent  of  500  feet  in 
length,  to  replace  by  a  simple  curve  of  1°  30'. 

93.  To  locate  a  tangent  to  a  given  curve  from  a  fixed 
point  without. 


Let  QTR  be  the  given  curve  and  P  the  point.  If  the  ground 
is  clear  and  the  point  not  over  200  feet  distant,  proceed  as 
follows  : 

Measure  in  the  direction  of  the  curve  to  Q,  and  onward  to 
R.  Then  by  geometry 


x  PQ.  (85) 

With  one  end  of  a  tape  pivoted  at  P,  observe  where  the 
length  PT  cuts  the  curve.  This  will  be  the  point  of  tangency. 
If  the  distance  is  greater,  measure  as  before  to  Q,  and  observe 
the  angle  which  the  chord  QR  makes  with  the  tangent  Q7,  at 
Q.  Thus  the  central  angle  becomes  known,  OM  and  QM  may 
be  calculated,  and  thence 


But 

and  the  angle 


tan  OPT  =  — , 
MPT  =  OPT-  0PM. 


90  MISCELLANEOUS    PROBLEMS. 

Hence  deflecting  at  P  the  angle  MPT,  the  direction  of  the 
tangent  is  indicated.  Its  length  is  given  in  Eq.  (85)  the  chord 
being  either  measured  or  computed. 

EXAMPLE.  — Given  QTR  a  4°  40'  curve,  P  a  point  1000  feet 
from  Q,  and  the  angle  1QR  18°  30',  to  find  the  angle  MPT 
and  the  length  of  the  tangent  PT. 

94.  To  locate  a  definite  point  in  a  given  curve  from  some 
point  in  the  tangent. 


Let  JK  be  a  tangent  to  an  8°  curve,  T  the  point  of  tangency. 
It  is  required  to  locate  from  some  point  in  the  tangent,  a  point 
P  in  the  curve,  two  stations  from  T. 

From  the  known  degree  of  curve  and  the  number  of  stations 
it  will  be  perceived  that  a  =  l  6°. 

Calculate  TP  =  2  R  sin  8°,  and  establish  P'  at  the  same 
distance  from  T.  Then  in  the  isosceles  triangle  PTP'  find 
P'P  =  2  PT  cos  TPM  =  2  PT  cos  4°. 

Set  up  the  instrument  at  P'  and  deflect  from  the  tangent,  in 
the  direction  of  the  curve,  one  fourth  the  central  angle  or  4°, 
and  measure  off  the  distance  PP'  to  the  point  P. 

95.  Given  the  perpendicular  distance  of  a  point  from 
a  tangent,  it  is  required  to  find  the  point  on  the  tangent 
whence  a  curve  with  a  given  radius  may  be  run  which 
shall  pass  through  the  given  point. 

Let  BT  be  the  tangent,  TP  =  d  the  perpendicular  to  the 
given  point  P,  OP  =  H  the  given  radius  ;  it  is  required  to 
find  x,  the  distance  from  T  to  the  point  of  curve  fi. 


TO   PASS    A    CURVE   THROUGH    A    POINT. 


91 


Draw  PE  parallel  to  HT ;  then,*in  the  right  triangle  POE, 
we  have 


x\ 


or 


(86) 


Given  7J7'  and  7V, 
the  student  may  write 
an  expression  for  7i. 

Show  how  either  of 
the  problems  in  this 
article  may  be  solved  by 
trigonometry. 


96  To  prolong  a  straight  line,  as  LN,  beyond  a  tree,  a 
building,  or  any  obstacle. 

FIRST  METHOD.  —  Set  up  the  instrument  at  any  point  of 
the  line,  as  TV,  and  deflect  sufficient  to  pass  the  obstacle  to  any 
point  1\  Measure  NP,  remove  to  7J,  deflect  to  0,  making  the 
angle  QJ'O  double  the  angle  at  TV. 


FIG.  56. 


Measure  PO  =  PN,  place  the  instrument  at  0,  observe 
7',  plunge  the  telescope  and  deflect  to  72,  so  that  SOU  =  },- 
OPQ,  the  telescope  will  then  be  in  the  prolongation  of  LN, 
and 

(87) 


SKCOND  METHOD.  —  Deflect  00°  from  the  direction  of  the 
line  at  TV,  measure  to  P  a  distance  sufficient  that  7*0,  making 
an  angle  of  60°  with  PN,  will  clear  the  obstacle.  Measure 


92 


MISCELLANEOUS    PROBLEMS. 


PO  =  PN,  and  turn  the  felescope  in  the  direction  of  O/t,  the 
prolongation  of  LN,  by  deflecting  60°  from  the  direction  of 
PO.  NO  is  evidently  equal  to  PO  =  PN. 


THIRD  METHOD.  —  Erect  a  perpendicular,  NK,  of  sufficient 
length  that  a  line  passing  through  A'  parallel  to  LA7"  will  clear 


the  obstacle  ;  run  KM ;  lay  off  MO  =  N K,  and  a  right  angle 
turned  from  MO  will  indicate  the  direction  of  LN,  or  its 
prolongation  OR. 


FIG  59. 

OTHERWISE,   if    a    stream    or   pond,  measure    a   base  line 
LP,  and  the  angles  at  L  and  J> ;  then,  by  the  law  of  sines, 

~  ~  X  sin  P 


LN  = 
~ 


sin  (L  +  P) 


(88) 


TO    FIND    THE   RADIUS    OF    A    TRACK. 


93 


97.    Given  a  railroad  track  on  a  curve  to  find  the  radius. 
On  the  curved  track 

M 

^\1V 


XY  take  any  point  L, 
and  measure  a  straight 
line  LN,  and  from  its 
middle  point  Q,  measure 
the  perpendicular  QM 
to  the  track.  Then  de- 
noting the  radius  MO 
by  R,  the  chord  LN  by 
2  c,  and  the  middle  ordi- 
nate  MQ  by  m,  we  have, 
from  a  well-known  prop- 
osition in  Geometry, 


Q 


FIG.  60 


whence 


m  :  c  =  c  :  2  R  —  m, 
c?  +  m* 


R  = 


2m 


(89) 


98.    To  locate  a  curve  parallel  to  a  given  curve  and  at  a 
definite  distance  from  it. 

Let  BCDE  be  the  given 
curve,  B'B",  E'E",  etc.,  points 
on  radial  lines  through  B,  E, 
etc.  To  locate  B'E'  parallel, 
and  at  a  given  distance  from 
the  first,  use  the  same  deflec- 
tion angle  and  find  the  length 
of  chord  E'D'  from  the  pro- 
portion 


(90) 


:  E'V, 

or,  denoting  R  +  EE'  by  jR', 

a  well-known  formula  will  give 

E'D'  =2R'  sin  E  OD  ; 
and  similarly  for  the  curve  B"E". 

EXAMPLE.  —  Given  BE,  a  4°  curve  of  three  stations,  to  find 
the  length  of  chord  required  to  run  in  the  curve  B'E'  60  feet 
distant. 


94 


MISCELLANEOUS    PROBLEMS. 


99.  To  connect  two  parallel  tangents  by  a  reversed  curve. 

FIRST  CASE.  —  Given  the  length  of  the  straight  line  con- 
necting the  tangent  points,  and  the  perpendicular  distance 
between  the  tangents,  to  find  one  of  the  equal  radii  which 
shall  unite  the  tangents  by  a  reversed  curve. 


FIG.  62. 


Denote  OD  =  DO'  by  R. 
«       TT  by  c. 
«       Q:r  =  perp.  dist.  by  d. 

Draw  OE  to  the  middle  of  TD,  then  the  triangles  TOE  and 
TQT  are  similar,  and  the  homologous  sides  give  the  proportion. 


or  R==f~'  <J1) 

The  student  may  show,  what  is  assumed  in  the  foregoing, 
namely,  that  the  point  of  reversed  curvature  D,  is  at  the  mid- 
dle of  TT. 

SECOND  CASE.  —  The  radii  unequal,  the  same  lines  TT  =  c, 
and  QT  —  d  given  as  before,  and  the  length  of  one  of  the 
radii  OT  =  R. 

The  student  may  find  the  unequal  chord  lengths  DT  and 
DT,  and  show  that  the  radius  0'T  =  R'  is  equal  to  the 
product  of  DT  and  DT  divided  by  2  QT,  or  denoting  DT  by  c' 
and  DT  by  c",  show  that 


REVERSED    CURVES. 


95 


Again,  since 


and 


7?'  =  — 
~  2d' 

R  _       c' 
~R'~  c  —  c' 


(92) 


R'  =  ^-.-R.  (93) 

Given  a  3°  curve  AB,  a  straight  line  TT'  intersecting  it  at 
71,  making  an  angle  of  40°  54'  with  the  tangent  TN ;  it  is 
required  to  find  the  reversing  point  P  whence  an  8°  curve  may 
be  run  terminating  in  the  given  straight  line  TT'. 

fo' 


Let  E  indicate  the  P.  T.  of  the  required  curve,  and  draw 
TC  and  ED  perpendicular,  and  OB  parallel  to  the  given  line 
TT.  Now  R  and  R'  are  known,  the  angle  CTO  =  NTT'  = 
40°  54'  whence  the  angle  COT  and  side  CT=  ED  can  be  found, 
hence  the  angles  TOP  and  0'  and  the  point  P  determined. 


CHAPTER   VI. 

CONSTRUCTION. 

A.     GENERAL   DIRECTIONS,    DEFINITIONS   AND 
PROBLEMS. 

100.  The  position  of  the  center  line  of  the  road  being  finally 
determined  upon;  its  place  indicated  by  stakes,  their  elevations 
taken,  the  profile  made,  and  grades  established,  the  next  thing 
in  order  is  to  build  the  road.  The  work  must  be  carefully  set 
out;  for  example,  stakes  must  be  set  for  excavations  and  em- 
bankments, and  for  culverts,  trestles,  etc.  The  precise  location 
and  elevation  of  bridges  and  tunnels,  if  any,  must  be  marked 
out;  the  amount  of  cutting  and  filling  necessary  to  reduce  the 
inequalities  of  the  ground  to  conform  to  the  grades  must  be 
ascertained;  the  kind,  quality,  and  quantity  of  materials  to  be 
used  in  the  construction,  their  most  economic  transportation, 
where,  when,  and  how  to  be  delivered.  These  and  innumer- 
able other  problems  and  questions,  present  themselves  to 
the  engineer  during  the  progress  of  the  work,  for  solution  or 
answer. 

To  proceed  advantageously  with  this  work,  especially  if  the 
line  is  of  considerable  length,  a  re-organization  of  the  engineer 
department  is  usually  affected,  the  chief  engineer  having  now 
division  and  resident  engineers  to  assist  him. 

The  chief,  as  before,  has  charge  of  the  work,  gives  general 
directions,  passes  upon  bids,  estimates,  etc.,  and  decides  numer- 
ous questionable  matters  referred  to  him  by  the  division 
engineer. 

A  division  engineer  is  placed  in  charge  of  several  miles 
of  the  line  in  which  there  are  a  number  of  residencies,  and 
to  him  the  resident  engineers  report.  From  these  reports 
monthly  estimates  are  made,  and  forwarded  to  the  chief  for 
examination  and  approval. 


DEFINITIONS    AND    PROBLEMS. 


97 


A  resident  engineer  has  charge  of  the  construction  of  a 
few  miles  of  the  road,  and  it  is  his  duty  to  personally  superin- 
tend it.  With  his  assistant  or  rodman,  he  will  show  grade, 
or  line,  perform  the  necessary  measurements  and  computations 
for  the  monthly  estimates,  and  make  the  required  report  to  his 
superior  officer. 

101.  A  cross-section  is  a  vertical  section  taken  at  right 
angles  to  the  vertical  plane  which  embraces  the  center  line  ; 
its  extreme  limits  to  the  right  and  left  depend  upon  the  width 
of  the  road-bed,  the  transverse  slope  of  ground,  the  side  slope 
of  cut  or  fill,  and  the  cut  at  the  center.     The  difference  in  the 
elevations  of  the  surface  and  grade  at  the  center  gives  the  cut 
at  that  point. 

A  cross-section  should  be  taken  at  every  regular  station,  and 
at  every  other  point  where  the  surface  of  the  ground  changes 
its  slope  perceptibly,  whether  at  the  center  or  near  the  place  of 
the  side  stakes,  so  that  data  may  be  had  to  calculate  closely 
the  amount  of  material  removed. 

102.  A  grade  point,  or  a  point  where  the  natural  surface  of 
the  ground  intersects  the  grade,  is  neither  in  cut  nor  fill ;  such 
a  point  is  discovered  by  setting  the  target  of  a  leveling  rod 
equal  to  the  difference  between  the  height  of  instrument  and 
elevation  of  grade,  and  having  the  rod  moved  around  until  a 
place  is  found  where  the  target  is  equal  in  height  to  the  line 
of  sight.     A  stake  marked  0,0,  should  be  set  at  such  a  point, 
and  its  position  noted  in  the  cross-section  book. 


FIG.  &t. 


103.  Given  the  elevations  of  two  points  A  and  B,  their 
distance  apart,  and  the  gradients,  to  find  some  point  P, 
where  the  grades  will  meet. 


98  CONSTRUCTION. 

Denote  the  horizontal  distance  of  AP  by  x. 
"  "  "  PB    "  y. 

"  "  "  AB   "  d. 

"  gradient  of  AP  by  «. 

"  "  PB   "  6. 

"  difference  in  elevation  of  .1  and  B  by  e. 

Then  x  -f  ?/  =  J,  and  «a:  +  by  =  e,  using  the  plus  sign  when 
the  grade  rises  from  P  to  B  and  the  minus  sign  when  the  grade 
falls  from  P  to  B. 

From  the  last  two  equations  the  values  of  x  and  y  may  be 
found,  and  the  point  P  thereby  located. 

For  example,  suppose  the  gradient  from  A  rises  .5  foot  per 
station  and  from  P  to  B  it  falls  .8  foot  per  station,  the  differ- 
ence in  elevation  of  A  and  B  2.8  feet,  B  being  lower  than  A, 
and  from  A  to  B  there  are  10  stations. 

Then  x  +  y  =  10 

i«+|y=    2.8, 

whence  x  =  4  and  ?/  =  6. 

Showing  the  point  P  to  be  four  stations  from  A. 

104.  To  find  where  a  grade  will  pass  from  cut  to  fill  or 
vice  versa,  the  slope  of  the  ground  being  uniform  between 
stations. 


r 

.41 


FIG.  65. 


1.  Given  the  cut  at  station  40  equal  to  a,  and  the  fill  at  41 
equal  to  b,  find  the  grade  point  P.  By  similar  triangles  we 
have 

a  :  x  =  b  :  100  —  x, 


whence  x  =  .  (94) 

a  +  b 

If  a  =  10,  and  b  =  4,  x  will  be  71.4'. 


VERTICAL    CURVES.  99 

2.  Given  cut  at  station  40  equal  to  a,  rise  of  grade  per 
station  equal  to  m,  and  the  slope  of  ground  n  to  1,  find  x,  the 
distance  to  the  grade  point  P. 


whence  x=     100yia    .  (95) 

mn  +  100 

If  a  =  8,  slope  of  ground  10  to  1,  and  rise  of  grade  per 
station  1.5',  x  will  =  G9.G'. 

3.  Find  an  expression  for  £,  assuming  the  grade  falls  b  ft. 
per  station;  the  other  conditions  as  in  example  2. 

105.  Vertical  Curves.  Where  two  grades  meet  an  angle  is 
formed,  and  it  is  necessary  to  lessen  the  grade  at  the  point  of 
meeting,  and  for  a  short  distance  both  ways  therefrom,  say 
from  100'  to  200'. 

FIRST  METHOD.  —  A  very  practical  and  generally  sufficiently 
accurate  method  to  round  off  two  grades  is  to  make,  on  profile 
paper,  a  drawing  of  the  grades  to  a  large  scale,  say  2  feet  or 
2£  feet  to  the  inch  vertically,  and  40  or  50  feet  to  the  inch 
horizontally;  then  fit  a  proper  curve  to  them  and  scale  off  every 
25'  or  50'  the  distances  to  points  therein. 

SECOND  METHOD  --  Let  MN  and  NO  represent  the  grades 
to  be  rounded  off.  Measure  equal  distances  in  both  directions 


from  the  point  where  the  grades  meet,  as  NP  =  NQ  equal  say 
150'  or  200',  and  connect  the  points  P  and  Q  by  a  straight 
line.  If  the  angle  of  intersection  of  the  grades  is  very  small, 
this  line  PQ,  with  a  little  rounding  up  at  its  extremities,  may 
be  taken  for  the  grade  required.  Then  the  ground  must  be  cut 
away  or  filled  up  to  it,  depending  upon  the  work  whether  it  is 
in  cut  or  fill.  By  repeating  the  operation,  that  is  to  say 


100  CONSTRUCTION. 

connecting  R  and  S,  points  100"  each  way  from  Q,  and  also  R' 
and  S',  equidistant  from  P,  a  nearer  approximation  to  a  curve 
will  be  attained,  and  in  surfacing  the  road  any  slight  angulari- 
ties at  the  points  of  meeting,  as  at  R  and  S,  can  be  removed. 

106.   Difference  in  elevation  of  the  rails  on  curves.     The 

centrifugal  force  F,  of  a  car  moving  in  a  curved  track  of 
radius  R,  and  velocity  v,  is  given  by  the  well-known  general 

,,2 

formula  in  Mechanics  F  =  «.     To  oppose  this  force  the  outside 

rail  on  curves  is  set  higher  and  the  inside  rail  lower  than  the 
grade  of  the  center  line. 

The  problem  is  to  find  the  difference  in  elevation  of  the 
rails,  so  that  as  a  car  moves  round  a  curve  on  the  inclined 
plane  thus  formed  by  the  unequal  elevation  of  the  track,  the 
action  of  gravity  to  draw  it  down  the  plane  will  just  equal  the 
centrifugal  force.  We  have  given  R,  v,  and  the  distance 
between  the  centers  of  the  rails  d,  to  find  the  difference  in 
height  h,  the  linear  dimensions  being  in  feet,  and  v  in  feet  per 
second.  The  action  of  gravity  upon  a  body  lying  on  an 
inclined  plane  varies  with  the  sine  of  the  angle  of  the  plane. 
The  component  of  gravity,  therefore,  that  opposes  the  cen- 
trifugal force  =  32|  sin  a,  a  being  the  angle  of  the  plane. 

Substituting  the  value  of  sine  a,  and  equating  the  forces  we 
obtain 

I        6d' 
whence  »  =  1£|.  (96) 

If  instead  of  v  =  feet  per  second,  we  write  F  =  miles   per 

3600 
hour  =  rQ      v  there  results 

22\26dF2    _. 06687 dF2  ™ 


Substituting  4'.9  for  d,  which  is  about  its  proper  value  for 
the  standard  gauge  of  4'  8£",  and  writing  for  the  radius  its 
equivalent  in  terms  of  the  degree  of  curve,  there  results 


ELEVATION    OF    OUTER    RAIL    ON   CURVES.      101 

Which  is  the  formula  employed    to   calculate    the    following 
table  :  — 


V 

DEGREE  OF  CURVE. 

1° 

2° 

.  3° 

4° 
.09 

5° 
.11 

6° 
.14 

7° 
.16 

8° 
.18 

10° 
.23 

20 

.02 

.05 

.07 

30 

.05 

.10 

.15 

.21 

.26 

.31 

.36 

.41 

.51 

40 

.09 

.18 

.27 

.37 

.46 

.55 

.64 

.73 

.91 

50 
60 

.14 

.29 

.43 

.57 

.71 

.87 

1.00 

.21 

.41 

.62 

.82 

1.03 

1.23 

In  calculating  the  height  in  any  given  case,  the  velocity 
assumed  should  be  that  of  the  train  of  the  highest  speed  which 
will  regularly  pass  around  the  curve,  since  if  the  centrifugal 
force  developed  be  not  thus  counteracted,  an  accident  might 
occur  from  the  excessive  pressure  of  the  flanges  of  the  wheels 
against  the  outside  rail.  On  the  other  hand,  the  flanges  of 
wheels  on  cars  running  at  a  lower  rate  of  speed  around  the 
same  curve  would  be  forced  by  gravity  against  the  inside  rail. 
The  effect  of  this  would  be  to  wear  off  the  inside  surface  of 
that  rail,  but  it  would  not  jeopardize  life  and  property  to  the 
extent  of  the  former  pressure,  if  the  outside  rail  were  not  prop- 
erly elevated ;  and  as  it  is  obviously  impracticable  to  guard 
against  both,  the  value  of  V  indicated  above  should  be  used. 
It  is  better,  however,  all  things  considered,  to  reduce  the  speed 
of  fast  trains  when  running  round  sharp  curves  than  to  elevate 
the  rail  unduly. 

With  regard  to  the  elevation  of  the  outer  rail,  the  practice 
among  engineers  is  not  precisely  uniform.  Some  think  £  inch 
elevation  per  degree  sufficient  for  speeds  up  to  50  miles  per 
hour.  Others  give  f  of  an  inch  per  degree  up  to  5°,  and  ^  inch 
per  degree  thence  up  to  10°, 


102  CONSTRUCTION. 

Another  rule  is  to  elevate  the  outer  rail 

1  inch      for  a  1  degree  curve, 

2  inches     "     2       «          " 

3  "          "3       "          " 
31     «         «     4       «          « 

4  «         "5       "          « 

41.       «  «       g          u  « 

and  slacken  speed  for  greater  curvature  rather  than  elevate  the 
rail  above  the  last  height.  It  is  evidently  unwise  to  lay  down 
a  specific  rule,  based  upon  the  degree  of  curve  simply,  since 
the  location  of  the  curve  will  in  practice  enter  the  problem  as 
a  factor.  When  the  curve  is  in  a  low  place  approached  by  a 
heavy  grade,  more  elevation  should  be  given  than  when  the 
curve  is  on  a  summit,  or  at  or  near  an  important  station,  or 
wherever  the  conditions  favor  an  easy  approach. 

The  table  shows  the  theoretical  requirements,  and  the 
engineer  must  exercise  good  judgment  in  its  application  as  in 
all  things  else. 

The  difference  in  elevation  of  the  rails  on  a  curve,  if 
transition  curves  are  not  used,  is  the  same  from  the  P.C.  to  the 
P.T.  From  each  of  these  points  it  is  diminished  gradually 
along  the  tangent  until  it  becomes  zero.  The  range  for  this 
distance  is  from  50  to  200  feet,  depending  upon  the  curvature. 
For  a  3°  or  4°  curve  the  distance  should  be  about  100  feet;  for 
an  8°  or  10°  curve  about  200  feet.  For  a  compound  curve  the 
average  of  the  elevations  due  to  its  branches  gives  the  proper 
difference  in  height  at  the  P.C.C. 

It  is  the  practice  on  most  roads  to  increase  the  gauge  on 
curves  varying  from  \"  to  £". 

107.  Inasmuch  as  it  requires  more  force  on  a  curved  track 
than  on  a  tangent  to  overcome  the  resistance  of  motion,  it 
is  customary  and  proper  to  make  the  grade  somewhat  less  on 
curves  than  on  tangents.  This  is  called  easing,  lessening,  or 
reducing  grades  on  curves.  The  rules  adopted  are  based  upon 
the  assumption  that  the  resistance  increases  with  the  curva- 
ture ;  that  is,  the  resistance  offered  by  a  curve  of  2°  is  twice 
that  of  a  1°  curve. 


SETTING    SLOPE    STAKES.  103 

On  the  Central  Pacific  Railroad,  in  the  Sierra  Xevada  Moun- 
tains, grades  were  lessened  for  curvature  from  2  to  2£  feet  per 
mile  per  degree,  on  curves  from  2°  up.  Or  on  an  8°  curve, 
the  grade  was  made  from  16  to  20  feet  per  mile  less  than  on 
the  tangents.  On  the  location  through  Weber  Canon,  on  the 
Union  Pacific  Railroad,  grades  were  reduced  T^  per  degree 
per  hundred  feet  ;  that  is,  if  a  maximum  grade  on  tangent 
was  one  foot  per  hundred  feet  on  an  8°  curve,  it  was  made 
1  —  T^  X  8  =  /^  of  a  foot,  or  a  reduction  of  about  13  feet 
per  mile.  Other  roads  use  y^  per  degree  per  hundred  feet, 
which,  for  an  8°  curve,  makes  a  reduction  of  about  21  feet 
per  mile. 

B.     SETTING   SLOPE   STAKES. 

108.  The  nature  of  the  ground  to  be  excavated  being 
known,  or  assumed  to  be  known,  the  ratio  of  the  side  slopes 
—  base  to  height  —  may  be  fixed  upon  ;  the  depth  of  cut  at 
center  being  computed,  and  the  width  of  road-bed  decided, 
sufficient  data  are  furnished  to  set  the  dope  stakes,  which  indi- 
cate the  limit  of  cut  or  fill. 


H 


-4  D 

FIG.  67. 

Suppose  the  ground  is  level  transversely,  and  that  ABHI 
represents  the  cross-section. 

Denote  the  width  of  road-bed  AB  by  «?, 

"         "    cut  CD,  at  center,  "    c, 

"       CI  or  CH,  the  distance  from  center 
to  place  of  slope  stake,  distance  out,  "    d, 

and  the  ratio  of  base  EM  to  height  MH    "    r  :  s. 

Then  d  =     +    c  °° 


104 


CONSTRUCTION. 


ordinarily  for  clay  cuts    r  :  s  =  3  :  2, 
or,  d  =  -  +  '- 


(100) 


If  the  ground  slopes  transversely,  let  ABFCE  represent 
the  section.  The  line  EF  may  be  straight,  or  it  may  be 
broken  at  C.  The  distance  out  on  the  right-hand  side  will 
evidently  be  greater  than  that  for  the  level  section,  while  that 
on  the  left  side  will  be  less.  In  all  cases,  if  the  inclination 
CF  of  the  ground  be  ascertained,  the  distance  out  to  F  can  be 
readily  calculated  as  follows  :  — 


Obtain  directly  from  the  given  slope  of  ground  the  depth  of 
cut  at  foot  of  slope,  as  at  B. 

Denote  this  cut,  the  line  BO  by  c', 

«  «  OP  «    8, 

PF  «    x, 

"       the  slope  of  ground,  horizontal  to  vertical  "    m  :  n, 
"        "    side  slopes  as  before  "    r  :  «. 

Then  —  g  =  x, 

8   c> 

and  -  o  =  c  +  #» 

r 


whence, 
and 


w   . 
a  =  —  H 

2 


ms  —  nr 

mrc' 


(101) 
(102) 


SETTING  SLOPE  STAKES.          105 

or  assuming  and  inserting  values  as  in  the  preceding,  namely 
r  :  a  =  3  :  2,  there  results, 


Measure  out  the  distance  d,  sight  a  rod  on  the  point  thus 
reached,  and  see  if  the  observed  and  computed  heights  agree 
quite  closely,  say  within  a  tenth  of  a  foot.  If  they  do  not, 
probably  the  ground  does  not  slope  uniformly  as  was  assumed, 
and  the  work  must  be  revised. 

A  stake  must  be  driven  at  the  point  F,  marked  with  figures 
indicating  the  depth  of  cut  there,  preceded  by  a  C  for  cut,  facing 
the  center  line  of  the  road,  and  a  record  made  in  the  cross 
section  book  of  the  distance  out  and  depth  of  side  cut.  In  case 
of  a  fill,  the  letter  F  should  be  substituted  for  C.  Some 
engineers  prefer  the  signs  -f-  and  —  for  cut  and  fill  respectively, 
but  the  letters  are  preferable. 

Evidently  the  same  general  formula  (102)  for  upper  side 
stake  in  cut,  will  answer  for  the  lower  side  stake  in  fill,  and 
if  the  ratio  of  side  slopes  is  the  same  in  both,  3  :  2,  the  particu- 
lar formula  (103)  may  be  used.  On  account  of  drainage,  the 
width  of  road-bed  must  be  from  4  to  6  feet  wider  in  cut  than 
in  fill. 

REMARK.  —  The  ratio  for  ordinary  earths  is  f  to  1;  for 
solid  rock,  \  to  1  ;  and  for  loose  rock,  and  sand  in  embank- 
ment, 1  to  1. 

For   distance   out,   lower  side   stake  in   cut,   we  use  the 

following  notation  : 

Denote  ET\)y&'. 

AQ  "   c", 

'    QT  «   a:', 

"        distance  out  "    d'. 

And  the  other  notation  as  above. 


Then  -  g'  =  x, 

m 


and 


CONSTRUCTION. 

• 


whence' 

and  <r  =     +  .  (105) 

2       7NS  4-  nr 

which  formula  will  answer  for  distance  out,  to  down-hill 
stake  in  cut,  or  up-hill  stake  in  fill,  and  may  be  modified  as 
(102)  when  the  ratio  of  the  side  slopes  is  determined.  If  r  :  s 
=  2:1, 

(106) 


EXAMPLES. 

1.    Given  width  of  road  bed  20  feet,  depth  of  center  cut  13', 
side  slopes  r  :  s  =  3  :  2  ;  slope  of  ground  m  :  n  =  10  :  1  .     Find  d. 


and  substituting  in  (103) 


The  result  shows  that  the  rod  reading  at  the  side  stake,  if 
the  ground  slopes  uniformly  from  center  out  10  :  1,  should  be 
3'.47  less  than  that  at  the  center. 

2.  Find  in  Example  1  the  distance  out  to  lower  side  stake, 
and   depth  of  side   cut.     State  what  the  rod  reading  should 
be. 

3.  Given,  width  of  road-bed,  26  feet  ;   side  slopes,  ±  to  1  ; 
surface  slope,  14  to  1  ;  center  cut,  16  feet.     Find  the  distance 
out  both  ways  from  center,  and  the  rod  readings. 

109.  When  the  surface  of  the  ground  cuts  the  road- 
bed, part  of  the  work  is  in  excavation  and  part  in  embank- 
ment ;  this  is  called  side-hill  work. 

In  this  case,  the  upper  side  can  evidently  be  determined 
as  before.  For  the  lowrer  side,  a  formula  can  be  readily 
deduced. 


SETTING    SLOPE    STAKES. 


107 


The  distance  DG,  and  hence  the  fill  A  0,  can  be  determined 
from  the  slope  of  the  ground.  It  remains  to  find  the  distance 
OQ,  which  shows  the  position  of  E. 


FIG.  69. 

Denote  A  O  by  c", 
OQ   «   8', 
"        QE   «  x, 
and  the  other  notation  as  before. 


Then 


V  —  x 

m6         ' 


and 


(107) 


whence  8'  =  —       — 

?ns  —  nr 

which,  added  to  half  width  of  road-bed,  gives  the  distance  out, 
and  it  will  be  perceived  that  (107)  is  analogous  to  (101),  as 
might  have  been  inferred. 

EXAMPLES. 

1.  Given  center  cut  2  feet,  slope  of  ground  4  to  1,  slope  of 
sides  3:  2,  width  of  road-bed  20  feet.     Find  distance  out  both 
ways  and  the  corresponding  heights.     Locate  also  the  grade 
point. 

2.  The  problem  shown  in  figure  70  may  be  solved  like  the 
preceding.     For  from  the  slopes  the  value  of  c"  can  be  found, 


108 


CONSTRUCTION. 


and  the  preceding  formula  will  give  the  proper  result.  Given 
the  center  cut  CD  =  4  feet,  width  of  road-bed  20  feet,  and  data 
as  shown  in  the  figure,  locate  the  grade  point,  and  side  stakes, 
and  state  rod  readings  thereat. 


FIG.  70. 

110.  A  compound  section,  or  one  in  which  different  mate- 
rials are  found  in  the  same  section,  as  rock  with  earth  super- 
imposed, may  be  staked  out  according  to  the  principles  already 
established  if  the  center  depths  and  slopes  of  the  materials  be 
known.     In  widening  an  old  cut,  or  making  new  excavations 
in  the  vicinity  of  old  workings,  such  information  may  be  sup- 
plied.    With    these    exceptions,  however,  in   the   majority  of 
cases  it  is  expedient  to  make  approximate  settings  of  stakes 
for  the  loose  earth,  and  when  this  is  removed,  rectify  and 
complete  the  work. 

111.  With  a  little  practice  in  setting  slope  stakes,  especially 
if  the  method  above  given  be  used,  in  which  the  judgment  of 
the  inexperienced  may  be  improved,  the  young  engineer  learns 
to  make  a  very  close  estimate  regarding  the  position  of  a  side 
stake,  and  when  he  is  confident  of  his  ability  to  do  this  he 
should  abandon  the  use  of  formulas  and  proceed  as  follows  : 
Estimate  the  rise  of  the  ground  from  the  center  stake  to  the 
place  where  the  side  stake  should  be  placed;  set  up  the  level  so 
as  to  take  a  rod  reading  on  both  these  points.     Observe  the 
center  first,  and  with  the  known  center  cut,  width  of  road  bed, 


SETTING    SLOPE    STAKES. 


109 


ratio  of  side  slopes,  and  assumed  rise  from  center,  calculate  the 
distance  out.  Observe  the  rod  on  this  point  and  if  it  agrees 
within  the  prescribed  limits,  say  a  tenth  of  a  foot,  drive  a  stake 
there;  if  not,  estimate  again,  and  profiting  by  the  knowledge 
obtained  during  the  preceding  effort,  one  or  two  more  trials 
should  be  sufficient,  and  less  time  consumed  than  by  the 
formula. 


FIG.  71. 

For  example  suppose  the  road  bed  20  feet;  center  cut  8  feet; 
side  slopes  3  to  2,  and  we  estimate  the  rise  from  the  center 
stake  to  the  place  where  the  side  stake  should  be  set  equal  to 
4  feet.  Then  the  distance  out  should  be  10  +  «  (8  +  4)  =28 
feet,  and  if  the  rod  reading  at  center  was  7  feet,  that  at  the 
side  should  be  3  feet.  Now  suppose  the  reading  should  be 
found  only  2  feet,  at  28  feet  distance  from  the  center.  Such  a 
result  would  show  the  ground  to  be  higher  at  P  than  it  was 
estimated,  and  therefore  the  position  of  the  stake  farther  out. 

AVe  perceive  that  an  additional  distance  of  three  halves  of  a 
foot  will  take  up  the  rise  of  one  foot,  which  is  the  difference 
between  the  estimated  and  observed  rise,  but  since  the  ground 
rises,  the  rod  reading  at  this  distance,  29.5  feet,  will  be  less 
than  2  feet,  and  therefore  a  little  greater  length  must  be  taken. 
Try  (7  =  30'  which  gives  a  rise  from  road  bed  of  f  (30  — 10) 
=  13.3',  and  a  corresponding  rod  equal  to  15' — 13.3'=  1.7'. 
Measure  out  this  distance,  sight  the  rod  and  see  if  it  does  not 
agree  within  a  tenth. 

Proceeding  in  a  similar  manner  to  set  the  down-hill  stake, 
\ve  may  estimate  that  the  ground  falls  2  feet  from  the  center 


110 


CONSTRUCTION. 


to  its  place,  making  the  distance  out,  19  feet,  and  supposing 
the  rod  reading  at  center  2',  that  at  19'  should  be  4'.  Sup- 
pose it  is  found  to  be  3.5' ;  this  shows  that  the  slope  of^the 
ground  is  not  as  great  as  it  was  estimated,  and  that  the  posi- 
tion of  the  stake  is  a  little  farther  out.  Now  we  perceive  that 
three-fourths  of  a  foot  more  horizontally  will  bring  the  side 
slope  half  a  foot  higher,  which  is  the  difference  between  the 
estimated  and  observed  rod  reading,  but  since  the  ground  falls 
in  the  direction  of  the  measurement,  we  do  not  require  quite 
a  half-foot  rise,  perhaps  a  tenth  less  ;  that  is,  instead  of  a 
reading  of  3.5'  feet,  we  should  expect  a  reading  of  about  3.6'. 
For  this  we  must  go  out  10  -f-  f  (8  -f  2  —  3.6)  =  19.6'  Measure 
out  this  distance,  observe  the  rod,  and  see  if  the  agreement  is 
not  sufficiently  close. 

If  the  rod  reading  at  19'  had  been  greater  than  four  feet, 
it  would  show  that  the  stake  should  not  be  as  far  as  19'  from 
the  center. 

From  the  foregoing  considerations  respecting  slope  staking 
we  can  write  the  following 

RULE. 

In  excavation,  if  the  observed  rod  reading  is  J  I 

L  greater j 

than  the  computed,  for  the  supposed  site  of  stake,  it  indicates 
that  the  true  position  of  the  stake  is  farther  J  t . 

In  embankment,  if  the  observed  rod  reading  is  J  I 

L greater j 

than  the  computed,  for  the  supposed  site  of  stake,  it  indicates  that 
the  true  position  of  the  stake  is  farther  -<  >  • 

LoutJ 


Sta. 

Eleva- 
tions. 

Grade. 

Left. 

Center. 

Right. 

Area. 

C.  Yds. 

Remarks. 

43 

44 

490.8 
499.7 

473.2 
474.3 

+  16.5 

+  17.6 
+  25.4 

+  20.5 
29.5 

+  31.7 

650.5 
1129 

3260 

Width  of  road- 
bed, 18'. 
Grade  rises  I'.l 
per  station. 
Slopes,  j|:  1. 

25.5 
+  20.7 

29.7 

40.7 

SETTING    SLOPE    STAKES.  Ill 

112.  The   above  table  exhibits  a   form   of   recording   the 
notes.     The  last  two  columns  will  be  explained  farther  on. 
The  first  two  columns  are  taken  from  the  level  notes  ;  the 
third  and  fifth  calculated  from  the  elevations   and   adopted 
grade  ;  and  the  fourth  and  sixth  supplied  on  the  field  during 
the  operation  of  setting  slope  stakes.     The  plus  sign  indicates 
a  cut  at  station  43  of  17'.6,  and  that  the  right  and  left  side 
stakes  for  that  station  are  respectively  20'. 5  and  16'.  5  above 
the  plane   of  the   road  bed.     A   minus   sign  correspondingly 
placed  would  indicate  a  fill.     The  denominators  of  the  frac- 
tional expressions  show  how  far  these  stakes  are  placed  hori- 
zontally from  the  center.     In  general,  the  numerators  of  the 
fractions,  found  in  the  notes,  exhibit  the  cuts  or  fills  and  the 
denominators  the  distances  out. 

If  the  surface  of  the  ground  transversely  is  such  that  it  can- 
not be  considered  level,  or  as  having  a  uniform  slope  to  the 
side  stake,  the  cross-section  party  must  ascertain  the  inequal- 
ities and  make  record  of  them  in  the  cross-section  book. 

113.  In  staking  out  the  work  allowance  should  be  made 
for  the  change  in  volume  of  the  material  to  be  removed  from 
excavation  to  embankment.     If  the  nature  of  this  material  is 
earthy  it  will  occupy  less  space  in  embankment  than  in  exca- 
vation, and  vice  versa  if  rocky. 

In  regard  to  the  shrinkage,  however,  much  depends  upon  the 
condition  as  well  as  the  composition  of  the  material,  and  also 
the  manner  in  which  it  is  placed  in  the  fill.  If  it  is  wet,  or 
frozen,  and  simply  shovelled  or  even  dumped  from  scrapers  a 
greater  allowance  should  be  made  than  if  it  is  dry;  or  if  the 
conditions  be  the  same,  and  carts  or  wagons  be  used  to  trans- 
fer the  material  to  a  long,  large  fill,  a  less  allowance  should  be 
made  because  the  earth  becomes  solidified  by  the  impact  of 
the  horses  and  loads.  Moreover  sandy  soils  will  not  shrink  as 
much  as  those  in  which  clay  preponderates. 

A  fair  average  for  the  shrinkage  is  taken  at  one-tenth,  Ta^. 
That  means  that  a  fill  which  is  to  be  13.5  feet  at  grade  must 
be  made  15  feet  high  at  first  so  as  to  allow  for  a  settlement  of 
1.5  feet ;  or  a  fill  finished  at  10  feet  will  settle  to  one  of  9  feet. 


112  CONSTRUCTION. 

Rock,  on  the  contrary,  when  broken  increases  in  bulk  ;  the 
increase  depending  upon  the  size  of  the  pieces,  being  greater 
for  small  pieces  than  for  large  ones.  A  fair  average  increase 
may  be  taken  at  two-thirds,  f  ,  or  3  cubic  yards  of  rock  in  cut 
will  make  5  cubic  yards  in  fill. 

C.     CALCULATING   THE   EARTH   WORK. 

114.  From  the  cross  section  book  we  now  obtain  data 
sufficient  to  determine  the  quantity  of  earth  to  be  removed,  or 
the  amount  of  cutting  and  filling. 

The  cross  sections  being  parallel,*  and  having  been  taken 
sufficiently  near  each  other  that  the  lines  connecting  the  cor- 
responding points  of  any  two  consecutive  cross-sections  may  be 
considered  straight,  and  the  sides  of  the  figure  planes,  the 
prismoidal  formula  will  give  the  exact  quantity  of  earth  in  the 
solid.  Or,  if  one  end-section  vanishes,  being  at  grade  as  at  the 
end  of  a  cut  or  side  hill  wrork,  there  will  be  a  wedge-shaped  mass, 
or  a  pyramid  formed;  but  since  this  formula  is  applicable  to 
the  wedge  or  pyramid,  it  may  also  be  used  in  these  cases  to 
determine  accurately  the  cubic  contents.  For  illustration, 
conceive  a  prism,  a  wedge,  and  a  pyramid  having  equal  bases 
and  altitudes;  and  let  b  denote  the  area  of  each  base  and  h 
the  common  altitude. 

Then  the  volume  of  the  prism  =  Hi, 


45  +  &). 
The  vol  ume  of  the  wedge  =  £  bht 


The  volume  of  the  pyramid  = 


It  will  be  perceived  that  6,  £  6,  and  *  b,  represent  respectively 
the  middle  area  of  the  prism,  wedge,  and  pyramid,  and  therefore 

*  Except  on  curves  where  a  correction  is  made.     See  Art.  118. 


CALCULATING  THE  EARTH  WORK.      113 

the  volume  of  either  of  these  solids,  or  any  combination  of  them 
may  be  expressed  by  the  following  equation,  known  as  the 

PRISMOIDAL    FORMULA. 

r*=£(A,+  4M+B)\  (108) 

In  which  V  denotes  the  volume, 

h        "         "    distance  between  the  ends, 
A  and  B        "         "    end  areas, 

and  M       "         "    area  of  section  midway  between 

the  ends  A  and  B. 

REMARK.  —  The  term  prismoid  usually  suggests  a  body  com- 
posed of  the  solids  just  named,  that  is,  one  having  six  plane 
surfaces  of  which  only  two  are  parallel,  yet  the  "  prismoidal 
formula  "  has  a  much  wider  application  as  was  first  shown  by 
Ellwood  Morris,  C.E.,  in  the  Journal  of  the  Franklin  Institute 
in  1840.  Trautwine  says:*  "It  embraces  all  parallelepipeds, 
pyramids,  prisms,  cylinders,  cones,  wedges,  etc.,  whether  regular 
or  irregular,  right  or  oblique,  together  with  their  frustums, 
when  cut  by  planes  parallel  to  their  bases;  in  a  word,  any  solid 
whatever,  which  has  two  parallel  ends,  connected  together  by  either 
plane  or  by  longitudinally  unwarped  surfaces." 

Gillespie  f  shows  that  if  the  surface  is  warped,  being  "  gene- 
rated by  a  straight  line  resting  on  the  two  straight  lines  which 
join  the  extremities  of  the  twro  end  sections,  and  moving 
parallel  to  their  planes  or  perpendicular  to  the  axis  of  the 
road,"  the  prismoidal  formula  will  give  the  correct  result. 

Or  if  the  natural  surface  is  generated  by  "  a  straight  line 
which  rests  on  the  two  end  sections  and  moving  on  them  in 
such  a  way  as  always  to  divide  them  proportionally  "  the  formula 
is  applicable. 

If  a  ridge  or  hollow  runs  obliquely  in  one  direction  across 
the  solid,  from  end  to  end,  and  its  position  determined  with 
sufficient  accuracy  that  the  area  of  the  mid-section  as  well  as 


*  Trautwine  on  Excavations  and  Embankments,  page  5. 
t  Gillespie,  Roads  and  Railroads,  pages  3G8-9. 


114 


CONSTRUCTION. 


that  of  the  end  may  be  computed,  the  prismoidal  formula  will 
still  hold. 

115.    Sectional  Areas.  —  To  calculate  the  contents,  we  must 
compute  the  end  areas  and  the  area  of  the  mid-section. 


D 
FIG.  72. 

If  the  ground  is  level  transversely,  the  area  is  evidently 
that  of  a  trapezoid,  having  for  bases  the  width  of  road-bed 
and  the  sum  of  the  distances  out,  and  for  height  the  center 
cut. 

Let  d  and  d'  denote  distance  out  to  right  and  left  respec- 
tively, w  the  width  of  road-bed,  and  c  the  center  cut. 


Then 


A  =  ^  (w  +  d  +  d') . 


(109) 


If  the  ground  slopes  uniformly  from  F  to  E  through  C, 
or  if  it  has  a  uniform  slope  from  either  C'  or  C"  to  E  and  F, 
CD  being  the  center  cut  in  the  first  case,  and  C'D  and  C"D 
center  cuts  in  the  last  cases. 


F 


In  either  case  two  triangles  may  be  formed,  having  for  bases 
AD  and  DB,  and  FG  and  EH  their  respective  heights,  and 
two  more  triangles  having  for  their  common  base  the  center 


CALCULATING  THE  EARTH  WORK.      115 

cut,  and  for  heights  the  distances  out.  Calling  h  and  li'  the 
right  and  left  side  heights  respectively,  and  using  the  notation 
above,  the  equation  for  the  area  may  be  written, 

A  ==  H?  (h  +  h')  +  c-  (d  +  d') .  (110) 

4  2 

The  area  of  a  section,  as  ABECF,  Fig.  73,  may  be  found 
without  using  the  center  cut,  simply  by  subtracting  from  the 
area  of  the  trapezoid  EFGH  the  area  of  the  triangles  DEH 
and  AFG. 

In  case  the  ground  is  irregular,  sufficient  measurements 
should  be  made,  so  that  the  section  may  be  divided  up  into 
triangles  and  trapezoids  and  its  area  thereby  computed. 

If  the  inequalities  of  the  surface  are  numerous,  it  is  gen- 
erally sufficiently  accurate  to  plot  the  section,  average  the 
inequalities  by  stretching  a  silk  thread  over  them,  scale  off 
the  necessary  distances  and  heights,  and  compute  the  area  by 
some  of  the  preceding  methods. 

The  middle  area  is  found  by  first  averaging  the  correspond- 
ing lines  of  the  end  sections,  thereby  obtaining  the  mid-section, 
and  then  in  the  usual  manner,  by  formula  (110)  compute  its 
area. 

For  example,  if  the  center  cuts  at  the  ends  A  and  B  are 
12'  and  8'  respectively, 

and  the  corresponding  distance 

out  to  the  right  34'    and  28', 

and  the  distance  out  to  the  left  23.5'   "     17.5', 
the  side  heights  on  the  right       16'      "     12', 
the  side  heights  on  the  left  9'      "       5'  ; 

then  the  mid-section  would  have  for  center  cut  10', 

"  "  "  distance  out  to  right  31', 

"  "  «       left     20.5', 

"  side  height  to  right  14', 

«  "  '         "       left       7', 

and  its  area 

=  ?2  (14  +  7)  +  ~  (31  +  20.5)  =  362.5  sq.  ft. 


116  CONSTRUCTION. 

116.  The  volume  of  the  section  referred  to  in  the  last  para- 
graph, supposing  the  distance  between  the  ends  .1  and  B  to  be 
100  feet,  and  the  width  of  road-bed  20  feet,  will  be  computed 
as  follows  :  — 


Area  of  end  A 

16      9 


=  ?2  (16  +  9)  +  1?  (34  +  23.5)  =  470  sq.  ft. 


Area  of  end  B 

f)f\  0 

=  ™  (12  +  5)  +  5  (28  +  17.5)    =267  sq.ft. 
4  2 

Area  of  mid-section  from  last  paragraph  362.5  sq.  ft. 

...  V  =  —  (470  +  4  X  362.5  +  267)  =  36450  cu.  ft. 
6 

or,  36450  +  27  =  1350  cu.  yds. 

The  area  of  the  cross-section  at  each  station,  when  computed, 
is  placed  in  the  column  of  areas  in  the  table  on  page  110,  and 
the  quantity  of  material  between  two  cross-sections  is  placed, 
as  shown,  in  the  column  of  cubic  yards. 

The  student  may  verify  the  areas  and  cubic  yards  in  the 
table  on  page  110. 

117.  Instead  of  the  prismoidal  formula  the  method  of  aver- 
aging end-areas  is  very  frequently  employed.  It  consists  sim- 
ply in  computing  the  areas  of  the  ends  of  the  sections,  taking 
their  arithmetical  mean,  and  multiplying  it  by  the  length  of 
the  section.  For  example,  suppose  we  take  a  mass  of  earth, 
level  on  top,  center  cut  at  one  end  12',  distance  out  each  way 
28',  the  road-bed  being  20'  and  sides  slopes  £  to  1  ;  the  cut 
at  the  other  end  14',  distance  out  each  way  31',  the  road-bed 
and  side  slope  as  before,  and  the  distance  between  ends  100'  ; 
then,  by  averaging  end  areas,  we  obtain, 

Area  first  end  =  (20  +  56)  1^  =    456  sq.  ft. 
"  second  "   =(20  +  62)11=    574    "    ' 


Average  =  =  515  sq.  ft, 


CALCULATING  THE  EARTH  WORK.     117 

and  the  volume  -  1907  cu.  yds. 

By  the  prisrnoidal  formula  the  correct  volume  is  found  to 
be  1904  cubic  yards,  but  more  labor  is  involved  in  the  calcula- 
tion. 

Again,  suppose  the  larger  end  section  to  remain  as  before, 
but  the  smaller  to  have  only  a  4'  cut  at  center,  the  width  of 
road-bed,  side  slopes,  and  length  of  section  remaining  un- 
changed, then  we  shall  find  by  averaging  end-areas  F=1255-f 
cubic  yards,  but  by  the  prismoidal  formula  the  correct 
quantity  is  found  to  be  1163  cubic  yards,  so  that  in  this  case, 
by  averaging  end-areas,  we  get  nearly  8  per  cent,  too  much. 
Furthermore,  if  the  smaller  end-area  vanishes,  the  surface 
being  at  grade  there,  the  other  dimensions  and  conditions  the 
same  as  before,  and  the  volume  be  computed  by  averaging 
end-areas  and  also  by  the  prismoidal  formula,  we  shall  find 
an  excess  by  the  former  method  over  the  exact  amount,  com- 
puted by  the  latter,  of  more  than  20  per  cent.  And  in  general 
it  will  be  found  that  in  a  prismoid  the  greater  the  difference 
between  the  end-areas,  the  greater  will  be  the  departure  from 
the  true  volume  when  the  method  of  averaging  end  areas  is 
used.  The  reason  becomes  obvious  upon  analyzing  the  figures 
found  in  the  various  cases.  In  the  first  and  second,  conceive 
a  plane  drawn  parallel  to  the  road-bed,  and  at  a  distance  from 
it  equal  to  the  center  cut  at  the  smaller  end.  Also  conceive 
two  other  planes  extending  throughout  the  section  longitudi- 
nally and  perpendicularly  to  the  road-bed,  one  through  each 
side  stake  at  the  smaller  end-section. 

These  planes,  with  the  surface  of  the  ground,  faces  of  the 
slopes,  and  the  end  sections  divide  the  solid  into  a  prism,  a 
wedge,  and  two  pyramids.  In  the  first  example  the  pyramids 
have  very  small  bases,  and  therefore  multiplying  as  we  did, 
when  averaging  end-areas,  by  one-half  the  altitude  instead  of 
one-third,  we  increased  the  volume  but  little.  In  the  second 
example,  however,  the  bases  of  the  pyramids  are  larger  and 
the  difference  in  the  product  between  one-half  and  one-third  is 
considerable. 


118 


CONSTRUCTION. 


In  the  third  or  last  example,  computed  above,  the  mass  may 
be  divided  into  a  wedge  and  two  pyramids,  the  bases  of  the 
latter  forming  a  still  larger  part  of  the  area  of  the  cross-section 
than  in  either  of  the  other  examples,  an  increased  percentage 
in  quantity  obtained  by  end-areas  over  the  correct  result  might 
be  expected.  While,  therefore,  in  many  cases  quantities  may 
be  computed  with  sufficient  accuracy  by  averaging  end-areas, 
it  will  be  perceived  from  the  foregoing  what  conditions  are 
favorable,  and  what  unfavorable  to  an  approximate  result  by 
this  method,  and  that  the  engineer  must  exercise  judgment  in 
determining  when  to  use  the  prismoidal  formula  instead. 
When  the  ground  is  very  much  broken  and  irregular,  or  in  the 
case  of  borrow  pits,  the  mass  may  be  divided  into  small  regular 
solids,  by  the  level  and  tape,  and  their  volumes  determined  by 
well-known  rules  of  geometry. 

When  the  precise  quantities  are  required,  as  in  piers,  abut- 
ments, etc.,  the  prismoidal  formula  should  be  used. 


118.    Excavation  on  Curves.  —  In  calculating  the  amount 
of  excavation,  we  have  thus  far  assumed  that  the  cross-sections 
are  perpendicular  to  a  straight  center  line.     The  assumption  is  j 
not  theoretically  correct  on  curves,  though  where  the  curve  is  j 


CALCULATING    THE    EARTH    WORK. 


119 


not  very  sharp,  the  error  arising  is  generally  slight.  Moreover, 
when  the  method  of  averaging  end  areas  is  employed  instead 
of  the  prismoidal  formula,  in  computing  quantities,  any  attempt 
to  correct  errors  arising  under  the  above  assumption  would 
be  a  needless  refinement ;  still,  where  greater  accuracy  is  de- 
manded, and  the  prismoidal  formula,  therefore,  freely  used  to 
obtain  the  volumes,  it  may  be  well  not  to  ignore  the  effect  of 
curvature,  especially  where  the  depth  of  cut  is  great,  the  radius 
small,  and  the  transverse  slope  steep. 

E 


Let  Fig.  74 x  represent  the  horizontal  projection  of  a  por- 
tion of  a  road-bed  in  excavation  on  a  curve,  the  center  of 
which  curve  is  at  0,  LCL'  the  center  line,  MEN  the  outside, 
and  UTV  the  inside  line  of  slope  stakes.  Now,  in  calculating 
the  volume  between  the  stations  L  and  C,  in  the  ordinary  way 
v/e  obtain  the  contents  of  the  solid  lying  between  the  planes 
PCQ  and  M'L  V,  drawn  at  right  angles  to  the  straight  line 
7,6',  thereby  getting  too  much  by  the  volumes  of  the  wedge- 
shaped  masses  QTC  and  LW,  and  too  little  by  the  wedge- 
shaped  masses  MM'L  and  PEC.  If  the  distances  out  were 
equal,  as  shown  in  cross-section  ABGCF  Fig.  74y,  the  over- 
Lipping  on  one  side  of  the  center  line  would  counterbalance 
the  gap  on  the  other,  and  no  correction  would  be  necessary. 
Iti  other  cases,  the  overlap  and  gap  may  in  general  be  repre- 
sented by  solids  similar  to  QCQ'  and  POP  respectively. 

Suppose  the  case  in  question  has  a  cross-section  as  A  BE  Fin 
74y.  We  may  proceed  as  follows,  employing  one  of  the  theorems 
of  Pappus,  namely  :  If  a  plane  area  revolve  round  any  axis  in 


120  CONSTRUCTION. 

% 

its  plane,  the  volume  generated  is  equal  to  the  area  of  the 
revolving  figure  multiplied  by  the  length  of  the  path  described 
by  its  center  of  gravity. 

Kow,  the  center  of  gravity  of  a  triangle  is  on  a  line  joining 
the  vertex  with  the  middle  of  the  base,  and  at  two-thirds  the 
distance  from  the  vertex. 

Let  H  indicate  the  middle  of  GE.  Draw  CE'  horizontally, 
and  project  //  and  E  on  it  at  H'  and  E'  respectively.  Calling 
the  distance  out  to  E,  d,  and  to  F,  d',  \ve  perceive  that 

-  \CH'  =  \ j.l(d-d')  +  d'  =  l(d  +  d') 
o  o    2  o 

or  £  the  sum  of  the  distances  out. 

This  added  to  the  known  radius,  will  give  the  radius  7t',  by 
which  we  may  determine  the  distance  traversed  by  the  center 
of  gravity  of  the  plane  in  question.* 

Let  RS  in  (x)  represent  the  required  distance  traversed, 
ES  =  G'S  -  G'R  =  G'S  -  100. 
TtR'D 


But  G'S  = 


180 


180 

Find  then  in  the  ordinary  way,  for  straight  center-line  by 
the  prismoidal  formula,  the  volume  of  the  mass  between  the 
stations  L  and  C.  From  the  mid-section  find  the  area  of  the 
triangle  CGE,  —  by  subtracting  from  the  whole  area,  already 
computed,  the  part  ABGCF,  —  which  we  may  assume  to  be  the 
plane  area  that  generates  the  solid,  or  the  wedge-shaped  body 
sought.  I 

Then  the  product  of  the  length  of  RS,  and  the  area  of  bGE 
•will  give  C  the  correction  required. 

The  correction  must  be  added  when  the  highest  ground  is  on 
the  convex  side,  and  substracted  when  the  highest  ground  is  on 
the  concave  side. 

*  Practically  correct,  but  not  precisely  so  theoretically. 


CALCULATING    THE    EARTH    WORK.  121 

EXAMPLE. 

Given  a  10°  curve,  or  72  =  573.7. 

Cross  section  at  A,    .     .     .     .    T4F  ,  T°¥,  f  §  ,  and  its  area  =  746 
"A     .     .     .     .    A,T%,|«,      "         "      =370 
"      M,     ....    A,  A,  If,      «         «      =545 
Vol.  between  the  planes 

PCQ  and  M'LV'  =  £f  (746  +  2180  +  370)  100  =  2035  cu.  yds. 
Area  of  triangle  CGE  in  74y  =  335 

CH'  =  i  (15  +  50)  =.21.67  . 

E'  =  573.7  +  21.67  =  595.4. 

3.1416X595.4X10 


= 


180 


Therefore  we  obtain  for  the  entire  volume  between  MV  and 
ET,  2035  +  48  =  2083  cu.  yds. 

119.    The  following  is  Henck's  method  of  making  the  cor- 
rection at  each  station.*     Adopting  the  notation  employed  in 
the  preceding  articles,  namely, 
c  =  the  cut  at  the  center, 

d  and  </',  the  greater  and  less  distances  out,  respectively, 
h  and  /*',  the  corresponding  side  heights, 
w  =  the  width  of  road  bed. 
Then  the  area  of  the  triangle 

CGE  =  ±c(d  —  d')  +  $  w  (h  —  hy 

The  wedge-shaped  mass  horizontally  projected  at  PCP'  is 
considered  a  truncated  prism,  its  edges  PPf  and  7t.S,  always 
short,  are  taken  as  straight  lines,  and  at  C  the  height  of  the 
.solid  vanishes. 

Now  PP'  =  2  d  sin  $  D, 

and  J2,S'  =  2  d'  sin  £  D. 

Then  since  the  volume  of  the  prism  is  equivalent  to  the 
product  of  the  base,  and  one-third  the  sum  of  the  edges,  the 

*  Henck's  Field  Book,  page  112. 


122  CONSTRUCTION. 

formula  for  the  volume  and  hence  the  correction  sought  is 

C  =  [$c(d  -  d')  +  ±w(h  -  h')]$(d  +  d')  sin  \D     (111) 

or,  writing  for  sin  ^  D,  its  value  in  terms  of  the  radius  R, 

i  on 

C  =  [i  c  (d  -  d')  +  i  w  (h  -  h')]  —  (d  +  d').         (112) 

In  side-hill  work  with  such  a  cross-section  as  sBE,  let  I 
denote  the  base  of  the  cut  rB,  Fig.  74  (#),  then  the  height  of 
the  solid  at  E  is  the  same  as  before,  or  2  d  sin  £  D,  the  height 
at  B  is  w  sin  $  D,  and  at  any  point  between  D  and  B  as  s,  the 
distance  of  which  from  the  center  is  £  w  —  ft,  the  height  is 
o  (i  w  —  ft)  sin  £  D  =  (w  —  2  ft)  sin  $  D.  Now  calling  b  the 
base  of  the  cutting,  the  area  of  the  cross-section  sBE  =  \  ft/, 
and  hence  the  correction 

=  i  M  i  (9  j+  w+  ,,,_o  &)  Sin  i  7), 

or  C  =  i  ft/i  §  (W  +  ?c  —  ft)  sin  £  D.  (113) 

When  the  excavation  lies  on  both  sides  of  the  center  lint 
having  a  cross-section  rBE,  its  area=£  ft/i,  the  height  of  the 
solid  at  E  and  B  will  be  as  in  the  preceding  paragrapl 
respecting  2  </  sin  £  D  and  w  sin  £  D;  but  at  a  point  betweei 
A  and  D  as  at  r,  the  distance  of  which  from  the  center  i; 
ft  —  ^  iv,  the  height  will  be 

2  (ft  —  i  M?)  sin  i  79  =  (2  ft  —  w)  sin  £  D. 

I 

This  last  height  being  on  the  opposite  side  of  the  center  lin< 
from  the  others  is  considered  negative  in  the  product  for  tin 
volume,  hence  the  correction 

=  i  bh  £  (2  d  +  w  —  2  ft  +  w)  sin  £  7)"; 
or  C  =  £  bh  §  (V  +  -a?  —  ft)  sin  £  D,  (114) 

precisely  the  same  value  of  C  given  in  (113). 

Substituting  for  sin  £  D  its  value  in  terms  of  the  radius  w 
obtain 

C  =  lbh™(,l+w-b-).  (115) 

6  K 

Add  or  subtract  as  indicated  above. 


CALCULATING    THE   EARTH    WORK. 


123 


EXAMPLE.  —  Given  the  road  bed  30  feet,  the  radius  of  the 
curve  800,  the  base  of  a  side-hill  cut  26,  distance  out  to  highest 
stake  60,  and  its  corresponding  height  28,  all  dimensions  in 
feet,  to  find  the  correction. 

C  =  i  X  26  X  28  x  2Yo°o  (G0  +  30  —  26)  =  971  cubic  feet- 

120.  Overhaul.  —  When  contracting  for  the  removal  of  ma- 
terial from  excavation  to  embankment,  it  is  sometimes  stipu- 
lated that  extra  pay  shall  be  had  for  the  transportation  of 
material  through  a  distance  greater  than  that  specified,  rang- 
ing usually  from  300  to  500  feet.  It  may  be  necessary,  there- 
fore, to  ascertain  what  is  this  extra  distance  known  as  overhaul, 
and  on  how  much  material  the  contractor  is  entitled  to  extra 
pay. 


FIG.  75. 

In  the  figure  let  the  straight  line  SPOR  represent  the  grade 
line,  and  the  curved  line  NOQR  the  profile  of  a  proposed  rail- 
road, where  OQR  is  to  be  cut  away  and  OTS  filled  up. 

With  a  little  computation  of  quantities,  based  upon  the 
cross-section  notes,  and  a  few  measurements  and  trials  on  the 
profile,  the  points  P  and  M  can  be  located,  so  that  the  cubic 
yards  in  MOQ  and  PON  will  be  approximately  equal --near 
enough  for  practical  purposes  —  while  the  limit  of  free  haul 
will  be  indicated  by  the  distance  MP. 

Now,  the  contractor  is  entitled  to  additional  compensation 
for  the  transport  of  all  material  from  MQR  to  NPST,  that  is 
to  say,  if  the  center  of  gravity  of  MQR  be  at  g  and  that  of 
NPST  at  (f,  then  the  extra  distance  which  this  material  is 
hauled  =  gtf  —  PM,  cr  eg  -f  c'y'.  With  a  few  trials,  and  a 
little  figuring  as  before,  a  tolerably  close  approximation  as  to 
quantity  and  distance  can  be  made.  If  greater  accuracy  is 
required,  the  center  of  gravity  of  each  mass  should  be  deter- 


124 


CONSTRUCTION. 


mined  by  the  principles  of  Mechanics.  This  distance,  usually 
reckoned  in  stations  of  100  feet  as  the  unit,  multiplied  by  the 
quantity  of  material  transported  and  by  the  price  agreed  upon 
for  the  overhaul,  will  give  the  amount  due  the  contractor. 

EXAMPLE.  —  If  the  limit  of  free  haul  be  400  feet,  the  dis- 
tance between  the  centers  of  gravity  of  cut  and  fill  be  1000 
feet,  the  price  one  and  a  quarter  cents  per  cubic  yard  per  sta- 
tion of  100  feet,  and  the  quantity  of  material,  exceeding  that 
of  free  haul,  transported  from  cut  to  fill  =  8000  cubic  yards  ; 
then  the  extra  pay 

P  =  (10  -  4)  80CO  X  £  =  $600. 

D.     CULVERTS,   BRIDGES,   AND   TUNNELS. 

121.  Culverts  are  used  for  the  passage  of  water  from  one 
side  of  the  road  to  the  other  under  the  track.  If  practicable, 
they  should  be  constructed  perpendicularly  to  the  center  line 
of  the  roadway.  They  are  set  out  by  driving  stakes  on  the 
center  line,  and  at  the  corners  or  angles  indicating  the  limits 
of  the  foundations  ;  and  on  each  stake  should  be  marked  the 
depth  required  to  be  dug.  In  the  note-book  there  should  be 
made  a  sketch  of  the  culvert,  accompanied  by  a  record  of  its 
dimensions,  the  amount  of  cutting  at  each  stake,  location  of 
reference  points,  etc. 


FIG.  76. 

The  length  of  a  box  culvert  cd  placed  at  right  angles  to  the 
roadway  may  be  found  as  follows  :  — 

Let  w  =  the  width  of  the  roadway, 
«    a  =  the  altitude  of  embankment, 

"    — :  1,  the  ratio  of  side  slopes, 
n 

and  h  =  height  of  culvert ; 


CULVERTS    AND    BRIDGES.  125 

then  its  length  l  =  w  +  '2-(a  —  h).  (1 1(5) 

n 

If  TO  :  n  =  3  :  2, 

then  l  =  w  +  3(a  —  h).  (117) 

For  a  20-foot  fill,  16-foot  roadway,  and  a  culvert  6  feet  high, 
slopes  |  :  1, 

I  =  10  +  3  (20  —  «)  =  58  feet. 

Searles  *  says  that  in  box  culverts  the  span  varies  from  2  to 
5  feet,  the  height  in  the  clear  from  2  to  6  feet,  the  thickness  of 
walls  from  3  to  4  feet,  the  thickness  of  cover  from  12  to  18 
inches,  and  its  length  at  least  2  feet  greater  than  the  span. 
Furthermore,  when  the  span  required  is  more  than  5  feet,  and 
the  embankment  too  high  to  warrant  carrying  the  walls  up  to 
grade  as  an  open  culvert,  an  arch  culvert  should  be  used.  The 
span  varies  from  6  to  20  feet ;  the  arch  is  a  semicircle,  the 
thickness  varying  from  10  or  12  inches  to  18  or  20  inches. 
The  height  of  abutments  to  the  springing  lines  varies  from  2 
to  10  feet,  the  thickness  at  the  springing  line  from  3  to  5  feet, 
and  at  the  base  from  3  to  6  feet,  the  back  of  the  abutment 
receiving  the  batter.  The  wing-walls  stand  at  an  angle  to 
30°  with  the  axis  of  the  culvert. 

122.  To  set  out  bridge  abutments  when  the  bridge  is  on 
a  tangent,  proceed  in  a  manner  similar  to  that  indicated  for 
culverts,  working  from  the  axis  of  the  roadway  to  locate  the 
center  line  of  the  foundation  for  the  main  part  of  the  abut- 
ment and  its  outside  limits,  and  then  the  direction  and  extent 
of  the  pit  for  the  wing-walls.  All  governing  points  to  be 
referenced,  care  being  taken  that  these  stakes  be  placed  where 
they  will  not  likely  be  disturbed  during  construction. 

When  the  bridge  is  on  a  curve,  and  especially  if  the  span  is 
considerable,  the  center  of  the  abutment  or  pier  should  not  be 
at  the  intersection  of  the  axis  of  the  bridge  with  either 
of  these  axis,  but  on  a  line  LN,  Fig.  77,  called  the  bridge.- 
chord. 

*  Field  Engineering,  page  209. 


126 


CONSTRUCTION. 


FIG.  77. 


The  bridge-chord  is  a 
line  midway  between 
the  tangent  T  at  the 
mid-span,  and  the  chord 
M  whose  arc  is  limited 
by  its  intersection  with 
the  axis  of  the  masonry 
represented  by  ah  and 
cd.  The  line  LN  is 
then  used  as  the  basis 
of  measurement,  whence 
the  limits  of  the  work 
b  are  determined. 

If  L  is  inaccessible  it 
may  be  located  as  fol- 
lows: From  some  point 
P  in  the  center  line  set 
off  PQ  perpendicular 


to  the  bridge-chord  ;   make 

PQ^RT—Lb 

=  R  (vers  POT — £  vers  iOT). 


and  then  will 


L  =  R  sin  POT— 


LN 


(118) 


The  point  L  may  then  be  found  by  measuring  direct  with  a 
tape,  or  if  this  is  impracticable,  it  may  be  indicated  by  the 
intersection  of  transit  lines. 

123.  Trestles  may  be  staked  out  by  locating  the  position  of 
the  center  of  each  bent,  and  then  measuring  -the  proper 
distance  right  and  left,  to  fix  the  limit  of  the  foundation  for 
the  sill.  If  pile  bents  are  used  stakes  should  be  driven,  if 
practicable,  to  indicate  the  position  of  the  piles.  If  a  swamp 
or  body  of  water  is  to  be  trestled,  and  the  line  is  straight, 
stakes  or  poles  a  few  feet  high  may  be  set  in  the  line  on  firm 
ground  on  each  side  of  the  water,  and  the  piles  ranged  in  with 
sufficient  accuracy  by  them.  The  location  of  each  additional 
bent  being  ascertained  by  measurement  from  the  one  immedi- 
ately preceding. 


CULVERTS,    BRIDGES    AND    TUNNELS.  127 

If  the  line  curves,  two  transits  may  be  employed  to  indicate 
by  the  intersection  of  their  lines  of  sight  the  place  for  each 
bent. 

The  bents  may  be  placed  12  or  15  feet  apart,  and  for  single 
track  are  usually  composed  of  a  sill,  cap,  two  vertical  posts,  two 
batter  posts,  and  two  braces,  running  diagonally  from  sill  to 
cap;  all  of  12  X  12  timber  except  the  batter  posts  which  may 
be  10  X  12,  and  the  braces  3  or  4  inches  by  10  or  12.  For 
double  track  roadway  some  modification  of  the  preceding 
is  necessary,  more  bracing  is  introduced,  and  for  greater 
height  than  28  or  30  feet  some  form  of  built-up  post  is 
employed,  the  posts  being  fastened  together,  thus  forming 
a  bent  throughout  the  series,  or  the  bents  may  be  braced 
in  pairs,  making  a  pier,  and  the  space  between  the  piers 
spanned  by  a  truss. 

High  wooden  trestles  are  not  as  common  in  this  country  as 
formerly,  having  been  largely  superseded  by  iron  structures. 


AC  D  B 

FiG.  78. 

124.  Tunnels.  —  Great  care  should  be  exercised  in  setting 
out  tunnels.  A  first-rate  instrument  in  good  adjustment 
should  be  employed  for  observing  direction  of  the  line,  and 
the  best  steel  tape  used  for  measuring  the  distance.  A  spring 
balance  should  be  used  with  the  tape  so  that  a  constant  ten- 
sion may  be  had,  the  measurements  being  made  between  plugs 
driven  specially  for  this  purpose,  and  the  readings  should  be 
corrected  for  temperature. 

If  possible,  some  point  should  be  selected  in  the  line  on  the 
summit  of  the  mountain  as  at  M,  whence  an  unobstructed 
view  each  way  down  the  mountain  may  be  obtained.  Here  a 


128  CONSTRUCTION. 

monument  should  be  erected,  whence  by  numerous  and  careful 
observations  made  at  different  times,  and  using  the  mean  re- 
sult, stations  A  and  B  in  the  vicinity  of  the  ends  of  the  tunnel 
should  be  established  precisely  in  a  line  which  includes  a  point 
on  the  monument  at  M,  from  which  the  direction  of  the  head- 
ings at  D  and  C  may  be  given. 

It  was  discovered  at  the  Musconetcong  Tunnel  that  the  best 
time  during  daylight  to  make  an  observation  was  just  after 
sunrise.  In  summer  it  was  quite  impossible  to  do  accurate 
work  during  the  middle  of  the  day.  A  round  iron  pole  of  one- 
half  inch  diameter,  painted  white  and  red  alternately,  answered 
well  for  a  sight  pole.  The  plummet  lamp,  however,  sighted 
on  a  calm  clear  night  gave  the  best  results. 

It  is  sometimes  necessary  to  establish  two  or  more  stations 
on  the  mountain.  In  the  alignment  of  the  Hoosac  Tunnel 
there  were  four  permanent  stations  used.  In  such  cases  the 
difficulty  of  obtaining  accurately  the  direction  to  drive  the 
headings  is  considerably  increased.  In  the  Mont  Cenis  Tun- 
nel an  extended  system  of  triangulation  was  resorted  to  in 
order  to  secure  precision  in  the  location  of  the  axis  of  the 
tunnel. 

In  locating  the  Musconetcong  Tunnel  the  first  method  above 
exhibited  was  employed  with  remarkable  results  ;  the  difference 
in  the  alignment  of  the  east  and  west  headings  was  only  four 
hundredths  of  a  foot. 

If  in  addition  to  driving  a  tunnel  from  its  ends,  work  is  to 
be  conducted  from  the  foot  of  a  shaft,  carefully  constructed 
apparatus  must  be  provided,  and  extraordinary  care  observed 
in  its  manipulation  to  transfer  accurately  the  direction  of  the 
line  from  the  surface  to  the  foot  of  the  shaft.  Various  devices 
have  been  employed  by  engineers  in  solving  this  difficult  prob- 
lem. The  principle  thing  is  to  suspend  from  two  points  pre- 
cisely in  the  line,  but  on  opposite  sides  of  the  shaft,  at  the 
surface,  two  plumb  lines  reaching  to  the  place  of  the  tunnel 
and  obtaining  thence  the  proper  direction  of  the  underground 
working. 

In  the  Hoosac  Tunnel  the  line  was  transferred  1000  feet 
down  a  shaft  with  such  precision  that  when  the  heading,  driven 


TUNNELS.  129 

over  two  miles  from  the  west  end,  met  the  one  produced  over 
2050  feet  from  the  plumb  lines,  the  error  in  the  alignment  was 
found  to  be  only  nine-sixteenths  of  an  inch,  and  the  heading 
driven  from  the  shaft  and  plumb  lines  1560  feet  in  the  oppo- 
site direction  met  the  corresponding  one  from  the  east  with  an 
error  of  only  Jive-sixteenths  of  an  inch. 

125.  In  running  the  levels  over  the  surface  corresponding 
care  must  be  ^exercised.  The  instrument  should  be  kept  in 
good  adjustment,  all  the  observations  made  on  benches,  and 
the  readings  taken  to  thousandths  of  a  foot.  The  levels  should 
be  repeatedly  tested  so  as  to  reduce  the  error  to  a  minimum. 

The  difference  of  levels  between  points  at  the  top  and  bottom 
of  a  shaft  may  be  obtained  by  measuring  with  a  rod  the 
distance  between  lines  made  on  a  number  of  bolts,  driven  from 
10  to  12  feet  apart,  in  a  vertical  line  down  the  shaft. 

The  grade  of  a  tunnel  should  be  at  least  sufficient  for 
drainage,  or  about  0.15  of  a  foot  per  station,  the  least  width  in 
the  clear  for  double  track  should  be  28  feet,  and  the  least 
height  in  the  clear  above  the  outside  rail  16  feet.  The  center 
of  the  tunnel  will  of  course  be  somewhat  higher,  depending 
upon  the  form  of  its  cross-section. 

Tunneling,  like  bridge  building,  has  become  an  engineering 
specialty,  and  one  who  desires  more  knowledge  on  the  subject 
should  consult  Drinker  and  Siinms. 

The  information  contained  in  the  following  extract  from 
Mr.  II.  8.  Drinker's  paper  on  the  Musconetcong  Tunnel  is  as 
important  to  the  engineer  studying  tunneling  to-day  as  when 
it  was  written  ;  it  is  thought  to  be  proper,  therefore,  to  give  it 
place  here.* 

The  approach  to  the  tunnel  on  the  west  begins  on  a  5°  curve, 
the  P.T.  of  which  is  about  800  feet  from  the  entrance,  and  the 
tunnel  itself  located  on  a  tangent  throughout  its  length,  the 
said  tangent  terminating  in  a  curve,  having  its  P.C.  some 
1850  feet  beyond  the  east  portal.  The  grade  ran  to  a  summit 


**Read  before  the  American  Institute  of  Mining  Engineers  at  New 
Haven,  Coun.,  Feb.  25,  1875. 


130  CONSTRUCTION. 

in  the  middle  of  the  tunnel,  the  same  being  the  summit  for  the 
road.  It  was  reached  by  a  rise  of  two-tenths  (0.2)  to  the 
hundred  feet  on  the  west  side,  or  10.56  feet  to  the  mile,  falling 
on  the  east  at  fifteen-hundredths  (0.15)  to  the  hundred  feet,  or 
7.92  feet  to  the  mile. 

To  determine  the  line  after  its  preliminary  location,  an 
observatory  was  erected  on  the  summit  of  the  mountain,  about 
12  feet  high,  with  an  eight-foot  square  base,  battering  on  the 
four  sides  about  1^  inches  to  the  foot.  T\vo  solid  stone  founda- 
tions were  also  built  on  line,  one  on  a  hill  about  half  a  mile 
from  the  west  entrance,  the  other  on  the  grading,  at  the  east 
end,  and  about  half  a  mile  from  the  eastern  portal.  As  the 
observatory  was  located  about  midway  over  the  tunnel,  this 
gave,  approximately,  equidistant  sights  of  about,  say,  a  mile 
and  a  quarter  each,  at  the  farthest.  This,  however,  was  done 
after  the  tunnel  had  been  started  from  points  established  on 
both  sides  by  repeated  and  carefully  checked  runnings.  The 
tower  being  subsequently  built,  two  points  were  established, 
one  each  on  the  foundations,  on  either  side,  from  the  lines  by 
which  the  work  at  either  end  had  been  so  far  run,  and  then 
assuming  these  end  points  as  correct,  by  a  series  of  repeated 
and  careful  trials,  the  center  point  on  the  tower,  or  permanent 
back-sight  for  both  ends  was  determined  by  setting  up,  approx- 
imately, over  it,  and  then  reversing  and  sighting  repeatedly, 
moving  the  instrument  to  and  fro  sideways,  within  a  variable 
distance  of  about  fifteen-hundredths  (0.15)  of  a  foot,  in  which 
the  sights  all  came,  and  finally  taking  their  mean.  This  was 
at  first  done,  as  soon  as  the  observatory  could  be  located  and 
built,  with  sufficient  accuracy  to  test  the  preliminary  lines. 
Subsequently  this  center  point  was  tested,  and  retested,  and 
determined  with  extreme  accuracy,  by  the  mean  of  very  many 
trials  made  both  by  sighting  by  day  and  by  night,  and  in 
winter  and  summer.  Different  objects  were  used  for  sighting 
on  in  day  work.  Both  the  ordinary  red  and  white  round  pole, 
also  a  flat  2x1  inch  black  pole,  with  a  white  center  streak. 
This  latter,  from  its  shape,  was  found  difficult  to  keep  plumb, 
either  when  held  or  fastened.  Also  a  pole  of  one-half  inch 
round  iron,  painted  white,  was  tried,  and  found  to  answer  well, 


TUNNELS. 

better  than  either  of  the  others.  But  far  better  and  more 
accurate  than  any  daylight  back-sight,  whether  permanent  or 
movable,  was  found  the  simple  expedient  of  using  plummet- 
lamps  on  clear  calm  nights.  They  worked  admirably  outside, 
a  flame  f  inch  high,  by  5  inch  in  diameter,  being  distinctly 
seen  in  the  long  sights  ;  and  with  a  fine  hair,  the  sights  were 
found,  finally,  to  repeatedly  test  within  practically  such  exact 
limits  (two  or  three  hundredths),  that,  the  point  being  once 
fixed,  it  was  not  subsequently  found  advisable  to  move  it. 
Now,  these  three  reference  points  being  located,  at  the  west 
end  a  center  was  set  at  the  mouth  of  the  slope,  and  from  it 
another  at  the  bottom.  This  gave  a  back-sight  of  276  feet  to 
run  from  into  the  heading.  At  each  shaft  a  center  was  first 
set,  with  great  care,  about  twenty-five  feet  off,  and  from  this 
the  line  prolonged  to  two  staples  driven  into  the  timbers  on 
each  side.  On  the  mean  of  many  sights  being  determined,  the 
points  on  both  staples  were  notched,  the  notches  tested,  and 
fine  plummet  lines  dropped  from  them,  the  weights  being 
steadied  at  the  bottom,  in  water.  Then  the  line  was  continued 
from  these,  as  in  ordinary  mine  surveying,  in  running  from  a 
shaft,  the  instrument  being  first  approximately  set  up  in  line, 
and  then  moved  sideways,  until  the  hair  exactly  bisected  the 
mean  of  the  slight  oscillations  observable  in  the  lines.  Though 
the  distance  to  be  run  from  the  shafts  was  not  great,  this  care 
was  necessary  from  the  shortness  of  the  back-sight,  the  distance 
between  staples  being  only  some  7£  feet,  and  from  the  fact 
that  the  headings  were  through  earth,  it  being  very  necessary 
i;i  enlarging  through  earth  to  be  able  to  have  the  crown  bars 
closely  located  at  equidistant  spaces  from  center.  On  the 
headings  between  the  shafts  and  slope  meeting,  the  various 
runnings  all  tested  closely  ;  but  it  was  the  long  line  between 
the  main  east  and  west  headings  that  required,  of  course,  the 
most  care,  and  caused  the  most  anxiety.  This  line,  at  the  east 
end,  was  simply  continued  on  the  grading,  up  into  the  heading, 
at  first  with  one,  and,  subsequently,  as  the  headings  advanced, 
with  two  intermediate  centers.  At  the  west  end,  the  line  was 
at  first  run  into  the  main  heading  (Xo.  1)  down  the  slope,  but 
as  the  enlargement  in  soft  ground  proceeded  between  the  slope 


132  CONSTRUCTION. 

and  west  end,  in  time  a  clear  sight  was  obtained  from  the 
mouth  of  the  tunnel  to  the  slope,  and  thence  into  the  heading, 
making  two  intermediate  centers,  as  at  the  east  end.  It  was 
always  necessary  to  have  a  station  where  the  slope  came  down, 
since  the  latter  was  driven,  after  meeting  rock,  sixteen  feet 
wide  —  thirteen  on  the  left  and  three  on  the  right  of  center 
line,  leaving  at  its  foot  about  ten  feet  of  space  for  passage  on 
the  right,  as  the  line  ran,  and,  of  course,  cutting  off  center  line. 
The  three  feet  on  the  right,  however,  were  dressed  off,  sub- 
sequently, at  the  level  of  the  heading,  so  as  to  give  a  clear 
back-sight  to  the  mouth. 

These  east  and  west  lines  were  repeatedly  run  and  tested 
as  the  headings  advanced,  and,  besides  the  work  continually 
spent  on  them  by  the  division  and  resident  engineers,  they 
were  frequently  checked  by  the  principal  assistant  engineer. 
They  finally  tested  within  four  hundredths  (0.04)  of  a  foot,  or 
less  than  one  half  an  inch.  The  levels  were  carried  over  the 
mountain  by  a  series  of  test  benches  run  until  succeeding 
benches  tested  within  0.005  of  a  foot.  On  meeting,  the  face 
benches  on  either  side  were  found  to  test  within  0.015  of  a 
foot,  or  less  than  one-fifth  of  an  inch.  Owing  to  the  system 
of  center  cuts,  used  in  blowing  the  rock,  in  which  ten  feet  at 
a  time  were  brought  out,  it  was  especially  necessary  that  the 
chaining  should  be  accurate,  so  that  the  distance  apart  of 
the  headings  might  be  safely  determined.  To  measure  over  the 
mountain,  two  stout  frames  were  made,  steadied  by  weights 
on  the  legs.  They  each  simply  consisted  of  a  vertical  shaft 
with  three  legs,  one  movable.  From  a  board  nailed  on  the 
top  of  the  shaft  a  fine  plummet  was  hung.  The  two  were  put 
in  line,  the  plummets  centered  by  the  transit,  and  a  point  at 
the  top  of  one  line  leveled  with  a  point  near  the  bottom  of 
the  other,  and  the  measurement  thereon  taken  between  the 
two  with  steel  tapes.  The  hind  frame  was  then  moved  on, 
and  the  chaining  so  carried  up  or  down  hill  in  successive  steps. 
This  method  was  found  to  be  satisfactory  ;  for,  on  the  head- 
ings coming  together,  the  distance  apart,  predicted  and  marked, 
was  found  to  agree  with  the  measured  distance  within  fifty-two 
hundredths  of  a  foot  (0.52),  or  about  six  inches  out  in  a  total 


TUNNELS.  133 

chaining  of  about  eight  thousand  feet,  —  four  thousand  through 
headings,  and  four  thousand  over  the  mountain,  —  the  test 
measurement  being  brought  down  the  slope  on  angle  instead 
of  in  at  the  west  entrance. 

126.  Ballast  Stakes  are  set  every  50  feet  at  the  proper  dis- 
tance transversely,  to  indicate  the  width  of  the  base  of  the 
ballast  and  by  their  tops  the  upper  surface.  The  depth  of  the 
ballast  is  about  18  or  20  inches,  or  it  is  12  or  14  inches  below 
the  lower  surface  of  the  tie.  The  center  line  of  the  track  must 
again  be  retraced,  stakes  driven  and  centered ;  on  curves  every 
50  or  25  feet,  depending  upon  the  curvature  ;  on  tangents 
every  200  feet. 


CHAPTER   VII. 

FROGS   AND    SWITCHES. 
DEFINITIONS. 

127.  The  gauge  of  a  track  is  the  distance  between  the  rails 
of  the  track.  It  is  measured  from  the  inside  of  the  rails  as 
from  A  to  B,  Fig.  79. 


a.  The   gauge  line   is   the    line  from  which  the  gauge   is 
measured.     It  is  used  instead  of  the  rail  in  these  calculations. 

b.  The  distance  between  tracks  is  the  perpendicular  dis- 
tance between  the  gauge  lines  of  the  tracks,  as  BA'> 

c.  A  turnout  is  used  to  connect  one  track  with  another,  as 
AB,  Fig.  80. 


FIG.  80. 


d.    A  crossover  is  used  to  pass  from  one  to  another  of  two 
parallel  tracks,  as  AB,  Fig.  81. 


FIG.  81. 


DEFINITIONS. 


e.  The  point  of  switch  is  the  point  at  \vhich  a  turnout  or 
crossover  begins,  as  the  point  A,  Fig.  80. 

f.  The  point  of  frog  is  the  point  at  which  the  gauge  lines 
of  two  rails  intersect,  as  at  C,  Fig.  80. 


FIG.  82. 

g.    The  frog  angle  is  the  angle  formed  by  th3  gauge  lines 
at  the  point  of  frog,  as  F  at  5,  Fig.  82. 

li.    The   number    of   a    frog    is    found   by  constructing    an 
isosceles       triangle 
upon  the   lines  en- 
closing     the     frog 

ir 


angle  and  dividing 
its    altitude    by  its     A 
base. 

If  A,  in  Fig.  83, 

be    the  frog    angle 

,         ,  D  FIG.  83. 

make    AB  =  AC, 

and  draw  AD  perpendicular  to  BC,  then 

the  number  of  frog  =  n 
If  AD  =  8  and  BC  ==  1,  then  n  =  8. 

i.  A  crossing  frog  is  formed  by  the  intersection  of  two 
rails  which  are  on  the  same  sides  of  their  respective  tracks, 
as  at  A  and  D,  Fig.  90.  The  frogs  at  A,  D,  B  and  E,  taken 
collectively,  are  sometimes  called  a  set  of  crossing  frogs. 


. 

BC 


136 


FROGS    AND    SWITCHES. 


k.  The  lead  L  is  the  distance  from  the  point  of  switch  to  the 
point  of  frog,  measured  on  the  chord  of  that  rail  of  the  turn- 
out which  passes  through  the  frog,  as  AB,  Fig.  82. 

L  The  radius  of  a  turnout  is  the  radius  of  the  gauge 
line  of  the  rail  which  passes  through  the  frog  as  OA,  OB,* 
Fig.  82. 

m.  The  radius  of  the  main  track  on  a  curve  is  the  radius 
of  the  gauge  line  of  the  rail  which  passes  through  the  frog.* 

n.  A  crossing  slip  is  an  arrangement  of  two  sets  of  switch 
rails  in  connection  with  a  set  of  crossing  frogs  by  which  two 
tracks,  which  cross  each  other,  are  connected,  as  AB,  ML,  in 
Fig.  94. 

PROBLEMS. 

128.  Given  the  angle  of  the  frog  F,  and  the  gauge  g,  of 
a  turnout  from  a  straight  track,  Fig.  84,  to  find  the  lead 
L  and  the  radius  R  of  the  turnout. 


In  the  right  triangle  ABE.  The  angle  A  =  £  F,  since  the 
angle  C  at  the  center  =  F,  and  the  angle  BAE  at  the  circum- 
ference subtends  the  same  chord  AB, 


*  This  definition  we  prefer  to  that  usually  given,  as  it  enables  us  to 
simplify  the  formulas. 


TURNOUT   FROM   A   STRAIGHT   TRACK.         137 

hence  A  B  =  -5fu  =  BE  cosec  ^1  , 

sm^l 

or  L  =  —  g—  =  gcosec$F.  (119) 


The  isosceles  triangle  A  CB  gives 


and  72  =  —  -  —  =  -  cosec  |  F.  (120) 

2 


EXAMPLE.  —  Given  the  frog  angle  =  7°  10',  and  the  gauge 
4.75  feet  ;  required  the  lead  and  the  radius  of  the  turnout. 

Ans.    Z  =  76/;  72  =  608.' 

129.  Given  the  radius  R,  and  the  gauge  g,  of  a  turnout 
from  a  straight  track,  to  find  the  lead  L  and  the  frog 
angle  F,  Pig.  84. 

The  right  triangle  *HCB  gives 


or  cosF  =  - — £;  (121) 

R 

and  the  isosceles  triangle  A  CB  gives 

' 

(122) 


EXAMPLES. 


1.  Given  the  radius  771.85,  and  the  gauge  4.75  ;  required 
the  lead  and  the  frog  angle. 

2.  Given  the  radius  1151.92,  and  the  gauge  4.75  ;  required 
the  lead  and  the  frog  angle. 


138 


FROGS    AND    SWITCHES. 


TABLE  FOR  TURNOUTS  FROM  A  STRAIGHT  TRACK.* 


No.  of 
Frog. 

Angle 
of  Frog. 

Lead 
of 
Turnout. 

Radius 
of 
Turnout.t 

Degree  of 
Curve  of 
Turnouts.! 

4 

14°  15'    0" 

38.15 

154.45 

,370  47/ 

5 

11°  25'  16" 

47.74 

239.9 

24°  04' 

6 

9°  31'  38" 

57.20 

344.4 

16°  42' 

7 

8°  10'  16" 

66.67 

467.9 

12°  17' 

8 

7°    9'  10" 

76.15 

610.4 

9°  24' 

9 

6°  21'  35" 

85.67 

771.85 

7°  26' 

10 

5°  43'  29" 

95.12 

952.4 

6°  01' 

11 

5°  12'  18" 

104.61 

1151.9 

4°  59' 

12 

4°  46'  19" 

114.10 

1370.4 

4°  11' 

13 

4°  24'  19" 

123.58 

1607.8 

3°  34' 

14 

4°    5'  28" 

133.07 

1864.4 

3°  04' 

15 

3°  49'  06" 

142.58 

2139.9 

2°  41' 

*  g  =  4.75. 

t  These  refer  to  the  rail  running  through  the  frog.  For  approximate 
degree  of  curve  of  turnout  from  a  curved  track,  use  degree  of  curve  of 
turnout  =  degree  of  curve  in  table  ±  degree  of  curve  of  main  track. 


130.  Given,  in  Pig.  85,  the  radius  CB  of  the^  main 
track  =  R,  and  the  frog  angle  F,  to  find  the  lead  L  and 
the  radius  R'  of  the  turnout  from  the  outside  of  the  main 
track. 

In  the  triangle  CAB,  A  —  B  =  F.  For  A  +  C'AB  =  180°, 
and  the  isosceles  triangle  C'AB  gives  C'A B  =  C'BA  and 
C'BA  +  B  +  CBE  =  1 80°.  Therefore,  since 

CBE  =  F,  A  +  C'AB  =  C'BA  +  B  +  F,  or  A  -  B  ==  F. 
Again 

CB-CA  :  CB  +  CA  =  tanl(A-B):tzn±(A  +  B), 


or,  substituting  values,  there  results, 

(2R- 


9 


(123) 


TURNOUT  PROM  A  CURVED  TRACK. 


139 


The  half  sum  of  A  and  B  being  thus  found,  and  £  (A  —  B) 

p 
being  equal  to  — ,  A  and  B  are  readily  determined. 


138 


FROGS    AND    SWITCHES. 


TABLE  FOR  TURNOUTS  FROM  A  STRAIGHT  TRACK.* 


No.  of 
Frog. 

Angle 
of  Frog. 

Lead 
of 
Turnout. 

Radius 
of 
Turnout.! 

Degree  of 
Curve  of 
Turnouts.  t 

4 

14°  15'    0" 

38.15 

154.45 

37°  47' 

5 

11°  25'  16" 

47.74 

239.9 

24°  04' 

6 

9°  31'  38" 

57.20 

344.4 

16°  42' 

7 

8°  10'  16" 

66.67 

467.9 

12°  17' 

8 

7°    9'  10" 

76.15 

610.4 

9°  24' 

9 

6°  21'  35" 

85.67 

771.85 

7°  26' 

10 

5°  43'  29" 

95.12 

952.4 

6°  01' 

11 

5°  12'  18" 

104.61 

1151.9 

4°  59' 

12 

4°  46'  19" 

114.10 

1370.4 

4°  11' 

13 

4°  24'  19" 

123.58 

1607.8 

3°  34' 

14 

4°    5'  28" 

133.07 

1864.4 

3°  04' 

15 

3°  49'  06" 

142.58 

2139.9 

2°  41' 

TABLE  FOR  TURNOUTS  FROM  A  STRAIGHT  TRACK. 

g  =  4'  8** 


No.  of 
Frog. 

Angle 
of  Frog. 

Lead 
of 

Turnout. 

Radius 
of 
Turnout. 

Degree  of 
Curve  of 
Turnout. 

4 

14°  15'    0" 

37.96 

153.0 

38°  09' 

5 

11°  25'  16" 

47.32 

237.8 

24°  17' 

6 

9°  31'  38" 

56.70 

341.4 

1(5°  51' 

7 

8°  10'  16" 

66.08 

463.8 

12°  23' 

8 

7°    9'  10" 

75.48 

605.0 

9°  29' 

9 

6°  21'  35" 

84.88 

765.1 

7°  30' 

10 

5°  43'  29" 

94.27 

943.7 

6°  04' 

11 

5°  12'  18" 

103.69 

1141.8 

5°  01' 

12 

4°  46'  19" 

113.10 

1358.4 

4°  13' 

13 

4°  24'  19" 

122.51 

1593.9 

3°  36' 

14 

4°    5'  28" 

131.91 

1847.8 

3°  06' 

15 

3°  49'  06" 

141.33 

2121.1 

2°  42' 

TURNOUT  FROM  A  CURVED  TRACK. 


139 


The  half  sum  of  A  and  B  being  thus  found,  and  £  (A  — 

p 
being  equal  to  — ,  A  and  B  are  readily  determined. 


Then  C  =  180  —  (A  +  B), 

and  the  exterior  angle 

CBE=C  +  C"   ^F, 
or  C'  =  F—C. 

In  the  triangle 


(124) 


smA 


sin  A 


(125) 


140 


FROGS    AND    SWITCHES. 


In  the  triangle  ABC', 
C'B  = 

or  R' 


AB 


2  sin  |  C" 
L 


(126) 


EXAMPLE.  —  In  a  turnout  from  the  outside  of  a  6°  curve 
with  a  number  10  frog,  find  the  lead,  and  radius  of  the 
turnout. 

131.  Given,  in  Fig.  85,  the  radius  CB  of  the  main  track 
=  /?,  and  the  radius  C'B  of  the  turnout  =  R't  to  find  the 
lead  AB  =  L  and  the  frog  angle  F. 


TURNOUT  FROM  A  CURVED  TRACK.     141 

Draw  ED  and  C'G  perpendicular  to  CH  and  AB  respec- 
tively. CAC/  is  a  straight  line,  for  the  curves  AM  and  AL 
are  tangent  to  each  other  at  A.  In  the  triangle  CBC',  CC' 
=  R  +  R'  —  g,  hence  all  the  sides  are  known,  and  we  have 
the  proportion 

CC'  :  BC  +  EC'  =  EC  —  EC'  :  CD  -  C'D, 
or  R  +  R'-g:  R  +  R'  =  R-R':  CD—  C'D. 


R  +  R'-g 

The  difference  between  CD  and  C'D  being  thus  found,  and 
having  their  sum  =  R  +  R'  —  g,  CD  and  C'D  are  readily 
determined. 

In  the  triangle  BCD, 


and  in  the  triangle  C'BD, 

—m-c-w-'       «** 

now  the  angle  CBE,  which  is  =  F  =  the  sum  of  the  angles  at 
C  and  C", 

or,  F=  C  +  C'.  (129) 

The  isosceles  triangle  AC'E  gives 

£  =  2E'siniC".  (130) 

EXAMPLE.  —  The  radius  of  a  turnout  from  the  outside  of 
a  4°  curve  =  1060.22.  Find  the  lead  and  frog  angle.* 

Ans.   Z  =  76.27  feet;  F=7°9/. 

132.  Given  the  frog  angle  F,  the  radius  CH=CB  =  R 
of  the  main  track,  Fig.  86,  to  find  the  lead  AB  =  L,  and 
the  radius  C'A  =  C'E  =  R'  of  the  turnout  from  the  inside 
of  a  curved  track. 

Let  AM  represent  the  outside  rail  of  the  main  track.  Then 
C,  C',  and  A  are  in  the  same  straight  line,  since  the  arcs  A  M 
and  AL  are  tangent  at  A. 

*  When  g  is  not  given  use  4.75. 


142 


FROGS    AND    SWITCHES. 


In  the  triangle  A  CB,  B  —  A  =  F.  For  F  being  the  angle 
between  the  tangents  at  B,  and  C'E  and  CB  being  radii,  it 
follows,  therefore,  that 


Then 


C'BC  =  F=B-  C'BA  =  B-A. 
F=B-  A. 


FIG.  86. 


Now  AC  —  BC  :  AC  +  BC  = 
hence,  tan*<4  +  B}  = 

or,  substituting,  there  results 


-  B)  :  t&nt(A  +  J5); 
-  B 


AC  —  BC 

tanJ-F 
9 


With  this  half  sum  of  A  and  B,  and  —  the  half  difference, 
A  and  B  may  be  found. 

Again,  C  =  180  -  (A  +  J5), 

BC  .  sin  C 


and 


AB  = 
L  = 


sin  A 
E  .  sin  C 


(131) 


TURNOUT  FROM  A  CURVED  TRACK.     143 

The  exterior  angle  at  C"  =  C  +  F,  and  the  isosceles  triangle 
A  C'B,  gives 

AC'  =      AB     , 
2  sin  4-  C' 

or,  ',  *'  =  _J,  =     coSeciC'.  (132) 


133.  In  Fig.  86,  given  the  radii  CB  =  R,  C'B  =  R',  to 
find  the  frog  angle  and  the  lead. 

In  the  triangle  CC'B,  CC'  =  CA  —  C'A  =  R  +  g  —  R'. 
The  three  sides  of  the  triangle  are  therefore  known,  and 
drawing  the  perpendicular  C'D,  we  have  from  a  well-known 
proposition, 

BD-CD  =  (BC'+CC')(BC'-CC>) 

BC 
or,  substituting,  there  results 


BD  —  CD  =  (2R'  —  R  —  g). 

R 

With  this  difference  of  BD  and  CD,  and  CB  =  R  as  their 
sum,  find  BD  and  CD.     Then,  since  the  angle  C'BC  =  F, 

^,  (133) 

-K 

and  cos  C'CB  =  CD  CD 


C'C      R  +  g  —  R' 
In  the  isosceles  triangle  A  C'B,  the  angle 

AC'S  =  C'BC  +  BCC'-, 

hence  A  B  =  2  C'B  sin  1 A  C'B, 

or,  L  =  2  R'  sin  1 A  C'B.  (134) 

EXAMPLES. 

1.  Given  the  frog  angle  =  5°  43'  29",  the  radius  of  the  main 
track  =  1436. 69,  to  find  the  lead  and  the  radius  of  the  turnout. 

2.  Given  the  radius  of  the  main  track  =  955.37,  and  that  of 
the  turnout  =  477.8,  to  find  the  frog  angle  and  the  lead. 

Ans.    F=  5°  43'  29";  L  =  94.83. 


144 


FROGS    AND    SWITCHES. 


134.  Given  the  angle  of  the  frogs  F=F/,  the  gauge  g, 
and  the  distance  between  the  tracks  b,  Fig.  87,  of  a  cross- 
over on  straight  tracks,  to  find  the  distance  F'K. 


H 


F'K 


FIG.  87. 
In  the  triangle  HFG,  the  angle  HFG  =  F,  and 


cosF 
In  the  triangle  HF/K^  the  angle  at  Ff  =  F,  and 


therefore  F'K  =  (b  —  g  sec  F)  cot  F.  (135) 

135.  Given  the  frog  angles  F  and  F',  the  gauge  g,  and 
the  distance  between  tracks  b,  Fig.  88,  of  a  crossover  on 
straight  tracks,  to  find  the  distance  CE  and  the  radius 

Make  DE  and  AB  perpendicular  to  BE,  and  A  G  parallel  to 
it.  Make  AC  =  g,  and  draw  A  K  parallel  to  the  tangent  of 

the  frog   at   C.     The 
L  D/^  angle  LA  G  =  F'. 

In       the       triangle 
KLD,  the  angle  LDK 
=  F,  and  the  exterior 
-    angle         KLM  =  F', 
hence  the  angle  LKD 

=  A  OD  =  F' F. 

In  the  triangle  A  CB, 
AC  =  g,  A=F',  AB 
=  AC  cos  A  =g  cos  F', 
and  BC  —  g  sin  F  . 
In  the  triangle  DGA,  DG  =  b  —  GE  =  b  —  g  cos  F',  and 
DA  G  =  LA  G  —  [LA D  =  ±(F'  —  F)]  =  £(F'  +  F), 


CROSSOVER  ON  CURVED  TRACKS. 


145 


AG  =  DG  cotDAG  =  (b  -  g  cosF')  cot  |  (F'  +  F), 
and       CE  =  BE-BC  =  AG  —  BC, 
or         CE  =  (b-gcosF')cot\(F'  +  F)-gs'mF'.  (136) 


and  the  radius 


DP 


sinDOP      sinKF'-F) 

substituting  the  value  of  AD  found  above,  we  obtain 
R==  _  b-gcosF'  _  . 

' 


.„ 


or 


R  =  b—9cosF'  cosec 


cosec 


—  F).  (137a) 


EXAMPLES. 

F'  ==  7°  9',  the  gauge 


4.75,  and 
4.75,  and 


1.  Given,  in  Fig.  87,  F 
6  =  7.417,  to  find  F'K. 

2.  Given,  in  Fig.  88,  F=-.7°9/,  F'  =  9°  32', 
I)  =  7.42,  to  find  CE  and  72. 

136.  Given  the  radius  FO  =  R  of  one  rail,  and  distance 
b  between  two  concentric  tracks,  and  the  angles  F  and  F' 
of  two  frogs  in  a  crossover  between  them,  Fig.  89,  to  find 
the  distance  FD  measured  on  a  chord  of  the  rail  BFD 
and  CO'  =  radius  of  the  outer  rail  of  the  crossover. 

Let  MC  and  BD  represent  the 
gauge  lines  of  the  rails  which 
pass  through  the  frogs  and  A  C, 
one  rail  of  the  crossover.  Draw 
the  radii  AO,  CO,  AO',  and  CO'; 
also  FO,  A  C,  and  FD. 

In  the  triangle  A  OF  the  ex- 
terior angle  OFO'  =  F,  A  F=y, 
and  FO  =  7i,  so  that  we  have 
two  sides  and  the  included  angle 
given,  whence 

R  +  g  :  R  —  g  =  tan  *  F  :  tan  -J  (OAF  —  A  OF), 


146  FHOGS    AND    SWITCHES. 


or 

g 

With  this  half  difference,   and  %  F  as  the  half  sum,  the 
angles  A  and  O  are  readily  found. 
Then,  by  the  law  of  sines, 

sinFAO  :  smAFO  =  FO  :  OA, 

or  0  A  =  RsmF  =  R  sin  p  cosec  FA  0.  (138) 

sinFAO 

In  the  triangle  AOC  we  now  know  OA,  OC  =  R  -j-  b,  and 
the  difference  of  the  angles  A  and  C=  (9J.jF-f-  .F',  for  the 
angle  OA  C  =  OA  F  +  O'A  C,  and  0  CA  =  O'A  C—0  CO'.  Now 
'  =  Fr,  but  (yCA  —  OCA=F'\  hence 


Then,  having  the  two  sides  AO  and  CO,  and  the  difference 
of  the  angles  opposite  them,  we  obtain,  by  the  law  of  tangents, 

(70+  AO  :  CO-AO  =  tanl(OAC  +  OCA)  :  tan$(OAC—  OCA); 

.-.tan  £  (04  C+  OCA)  =  CO  +  AOtanl(OAC-  OCA), 
CO  —  AO 

or         tani(OyK?+  OCA)=R  +  b  +  A0  tznl(F'  +  OAF). 

R  ~\~  b  —  A  0 

With  this  half  sum  and  —  —  as  the  half  difference,  we 

find  OA  C  =  their  sum  and  OCA  their  difference.     Then,  by 
the  law  of  sines, 


sin  OAC 


In  the  triangle  AO'C,  0/AC=0/CA  =  OCA  +  O'CO  and 
.40'C^ISO0  —  2  O'CA,  there  are  then  known  all  the  angles 
and  the  side  AC,  so  that 

/  (140) 


sin^O'C 
In  the  isosceles  triangle  FOB,  FOD  =  BOD  —  EOF,  and 


or  FD  =  2RsmiFOD.  (141) 


CROSSING    FROGS. 


147 


It  is  evident  that  F —  F'  must  be  a  small  angle,  since,  if  it 
were  not,  R'  would  be  too  small  for  practical  purposes. 

EXAMPLE.  —  Given  F=  7°  9'  10",  F'  =  5°  43'  29",  the  gauge 
4.75,  and  b  =  7.417,  to  find  R'  and  FD. 

137.  Given  the  angle  of  the  crossing  frogs  =  F,  and  the 
gauges  ff  and  g'  of  two  straight  tracks,  Fig.  90.  to  find 
the  distances  EA  =  DB  and  AB 


FIG.  90. 

Draw  HA  and  EC  perpendicular  respectively  to  ED  and 
AC.  On  account  of  the  parallelism  of  the  lines  ED  and  AB, 
and  of  DB  and  A  C,  the  angles  at  E,  D,  A ,  and  C  are  equal 
to  F.  In  the  right  triangle  ABC,  BC  =  g,  and  BAC=F, 

hence, 


AB=          =  q  cosec  F. 
sinF 


Similarly,  the  right  triangle  EAH  gives 


EA  =  -3—  =  0'  cosec  F. 
sin  ,P 


(142) 


(143) 


138.  Given  the  radius  EC,  of  a  rail  of  a  curved  track, 
Pig.  91,  and  one  angle  F,  at  E,  made  by  a  straight  track 
crossing  it.  Required  the  angles  F',  F",  F'"  situated  at 
D,  A,  and  B  respectively,  and  the  distances  EA  DB  ED 
and  AB 


148 


FROGS    AND    SWITCHES. 


In  the  triangle  EDO  the  angle  E  =  90  +  F,  the  angle  at 
=  9Q  —  F'EC  =  Ra,ndDC  =  R        .     Then 


.    n_ 


sinJ0  X 


DC 


sin  (90  +  F)R 
R  +  g 


or 
and 


R 


cosF, 


F'  =  90  —  JD. 
The  angle  ECD  =  F'  —  F,  and 

D sin  ECD  X  EC R  sin  (ffx  —  F) 

sinEDC 


(144) 


(145) 


FIG.  91. 
The  right  triangle  EHG  gives 

~~  cosGEII     cosF' 

In  the  triangle  CGA,  CA  =  R,   CG  =  R  —  EG  and  angle 
CGA  =  90  +  F.     Then 


sin  CGA  X  CG 


J2 


Now 


(146) 


CROSSING    FROGS.  149 

The  angle  EC  A  =  F"  —  F. 
The  isosceles  triangle  A  CE,  gives 

EA  =  EC  X  2sin$ACE, 
or  EA  =  2  R  sin  i  (F"  —  F).  (147) 

The  triangle  A  CB  gives 


sin  ABC  =  *™CABX  AC  =  _R_  cosj^ 
CB  R  +  g 

and  F'"  =  90-^4  BC.  (148) 

The  angle  A  CB  =  F"f  —  F", 

A  T,      sinACB  X  BC 


AB  =  -,  (149) 

cosjF1" 

In  the  triangle  BCD  the  angle  C  =  F"'  —  F',  and 


or  DjB  =  2  (R  +  0)  sin  £  (¥'"  —  F').  (150) 

EXAMPLES. 

1.  Given,  in  Fig.  90,  the  angle  F  =  9°  31'  38"  (No.  6  frog) 
to  find  EA  and  ED. 

2.  Given,  in  Fig.  91,   a  4°  curve,  and  a  No.  6  frog  at  E, 
required  F',  F",  F"',  and  distances  EA,  DB,  ED,  and  AB. 

139.  Given  the  radii  A  C  =  EC  =  R,  and  A  C'  =  BC'  = 
R',  of  two  curved  tracks  crossing  each  other,  Fig.  92,  and 
forming  the  angle  F  at  the  point  E.  Required  the  angles 
F'j  F",  and  F'"  ,  formed  at  A,  D,  and  B,  respectively,  and 
the  distances  EA,  ED,  AB,  and  DB. 

In  the  triangle  CEC'  we  have  given  CE  =  R,  C'E  =  R' 
+  g',  and  the  included  angle  CEC'  =  F.  Then 


tan*  (ECC-  -  EC'C)  =  (EC'  -  BC)i^(EQC'  +  EC'C) 

EC'  +  EC 


R'  +  g'  +  R 


150 


FROGS    AND    SWITCHES. 


With  this  half  difference  and 


180° 


-,  as  the  half  sum, 


the  angles  ECC'  and  EC'C  are  readily  found. 
This  triangle  also  gives 

ECsinCEC'    _    RsinF 


Now,  in  the  triangle  A  CC',  CC'  is  given  by  the  last  equa- 
tion, the  side  A  C  =  R,  AC'  =  R',  hence  the  angles  may  be 
computed. 

Draw  A  H  perpendicular  to  CC',  then 

C'H  —  CH  :  G'A  —  CA  =  C'A  +  CA  :  CC", 


-CH  = 


CC'  CC' 

This  half  difference  of  the  segments  of  the  base  added  to 
their  half  sum  will  give  the  longer  segment  C'H,  and  being 
subtracted  from  the  half  sum  will  give  the  shorter  CH. 
Then 


CA  C'  A 

and  1?"  =  1800  —  (^CC'  +  AC'C).  '          (151) 

Now,  in  the  isosceles  triangle  A  CE,  ECA  =  ECC'  —  ACC', 
and  CA  =  CE  =  R,  and  the  included  angle  C  are  known, 
hence 

EA=2Rsin$ACE.  (152) 


CROSSING    FROGS.  151 

In   the   triangle    CDC',   CD  =  R  +  //,    C'D  =  A'  +  </',   and 
CC"  is  known,  hence 

nw       r>' it'       (K  +  9)2  ~~  0 
u/z    —  Lf  jl  = « *• 


and 


and                     CDC'  =  F"  =  180°  -  (DO//'  +  DC'  II').  (153) 
Again,  the  angle  DC'C  —  EC'C=EC'D. 
In  the  isosceles  triangle  C'DE, 

C'E  =  C'D  =  Rf  +  <j,  and  ED  =  2(R'  +  g')  sin  |  #C'D.  (1  54) 


In  the  triangle   CZJC",  CB  =  R  +  y,   C'B  =  AJ/,  and   CC"  is 
known,  hence 


Proceeding  as  above,  with  the  half  difference  and  half  sum 
we  obtain  C'H"  and  CH",  then 


C£        B+flr 

cos  BC'  C  =  C/H//  =  G'H" , 
C',8          R' 

and  F'"  =  C7^C7  =  180  -  (BCC'  +  BC'C).    (155) 

In  the  isosceles  triangle  AC'S,  AC"  =  BC"  —  R',  and  the 
angle  A  C'B  =  BC'C  —  A  C'C,  hence 

^4  7J  =  2  7i"  sin  i  ^.  C'7^.  ( 1 56) 

In  the  isosceles  triangle  DCR, 

DCB  =  DCC'  —  BCC', 
and  DB  =  2(R  +  g)smiDCB.  (157) 

EXAMPLE.  —  Given,  in  Fig.  92,  a  1°  curved  track,  crossing 
another  of  4°,  and  a  number  6  frog  at  E.  Find  F',  F",  F'", 
and  the  chords  EA,  ED,  AB,  and  DB. 


152 


FROGS    AND    SWITCHES. 


140.  If  the  tracks  cross  as  in  Fig.  93  then  the  solution 
is  the  same,  except  that  CEC'  =  180°—  F,  and  F=ECC' 
+  EC'C. 

n 


The  half  sum  of  ECC'  and  EC'C =  — ,  and  (151)  becomes 

F'  =  A  CC'  +  A  C'C ;  (153)  becomes  F"  =  Z>C7/'  +  DC'H' ; 
(155)  becomes  Fx//  =  £CC"  -f  £C'C. 

EXAMPLE. —  Given,  Fig.  93,  a  1°  curved  track,  crossing  an- 
other of  2°,  and  a  No.  8  frog  at  E,  to  find  JF',  F",  F'",  and 
the  chords  ED,  EA,  ED,  and  EA. 

141.  Given  F,  the  angle  of  intersection  of  two  straight 
tracks,  Fig.  94,  to  find  the  radii  A  0  and  MO,  and  the 
lengths  of  the  curved  rails  AD  and  ML,  of  a  crossing  slip 
connecting  the  tracks. 

Draw  the  radii  AO  and  BO,  and  connect  C  and  .0.  By 
Article  137,  Fig.  90,  find  the  distances  GC,  GK,  and  C7/, 
HK,  and  assume  6V1  as  small  as  the  construction  of  the  frog 
at  G  will  permit. 

Then,  since  the  arc  ADD  must  be  tangent  to  GC  and  CH 
at  J.  and  D  respectively,  AC=  CB  =  GC —  GA.  F  is  the 


CROSSING    SLIPS. 


153 


angle  at  the  vertex,  and  A  C  the  tangent  distance  of  the  curve 

A  DB.     Hence 

AO  =  ACcot±F.  (158) 

MO  =  AO  —  g.  (159) 

j£ 


The  length  of  the  arc 


and  the  arc 


AB  =  AO  x  3.1410  X  -£- 


ML  =  MO  X  3.1410  X  ~ 
oOu 


(160) 


(161) 


EXAMPLE.  —  Given,  in  Fig.  94,  F=  9°  31'  38",  to  find  the 
radii  and  lengths  of  the  curved  rails  AB  and  ML,  GA  being 
5  feet. 

142.  In  a  crossing  of  a  curved  track  by  a  straight 
track,  Fig.  95.  Having  given  the  radii  of  the  rails  of  the 
curved  track,  and  the  angle  of  the  frog  at  E,  to  find  A  0', 
MO',  the  radii,  and  the  lengths  A  B  and  MN  of  the  curved 
rails  of  a  crossing  slip  connecting  the  tracks. 

By  Article  138,  Fig.  91,  find  the  angles  F',  F",  and  F'",  of 
the  frogs  at  C,  K,  and  //respectively,  and  the  distances  EC, 
EK,  CH,  and  HK. 


154 


FROGS    AND    SWITCHES. 


Draw  the  radii  EO,  CO,  BO,  and  HO.  Assume  BH  as  short 
as  the  construction  of  the  frog  at  H  will  admit  of.  Draw  DB 
tangent  to  CBH,  and  AO*  at  right  angles  to  ED,  making 


FIG.  95. 


In  the  isosceles  triangle  BOH, 


BH 


the  angle 

COB  =  COH  -  BOH  =  F"  -F'  -  BOH, 

and  the  chord 

CB  =  2  sin  i  COB  X  CO.  (162) 

In  the  triangle  DCB,  CBD  =  $  COB,  DCB  =  F'  +  $  COB, 
and  the  exterior  angle  FDB  =  their  sum  =  F'-\-  COB*  Then, 
by  the  law  of  sines,  CD  and  DB  are  readily  found.  It  will 
now  be  seen  that  FDB  is  the  angle  at  the  vertex,  and  DE 
the  tangent  distance  of  the  curve  AB.  Then 

=  BO'  =  BDcotiFDB.  (163) 

=  BO'-g.  (164) 


*  When  the  slip  rails  are  on  the  outside  of  the  curve  FOB  =F'  —  l  COB, 
and  FDB  =  F'—  COB. 


CROSSING    SLIPS. 


155 


The  length  of  the  arc 

AO'  x  3.1416 

360 
The  length  of  the  arc 

MO'  X  3.1416 
360 


X  the  angle  FDB  in  degrees.     (165) 


X  the  angle  FDB  in  degrees.    (166) 


EXAMPLE.  —  Given,  Fig.  95,  the  curved  track  on  a  2°  curve, 
and  the  frog  at  E,  a  No.  6  frog,  to  find  the  radii  and  lengths 
of  the  curved  rails  AB  and  MN,  BH  being  4|  feet. 

143.  Given  the  radii  of  the  rails  of  two  curved  tracks 
which  cross  each  other,  as  in  Fig.  96,  the  angle  of  the 
frog  at  G'  =  F,  and  AG,  to  find  the  radii  and  lengths  of 
the  curved  rails  AB  and  EF,  of  a -crossing  slip  connecting 
the  tracks. 


0  and  0'  being  the  centers  of  the  rails  GC  and  GF  respec- 
tively, and  the  figure  completed  as  shown,  0"  will  be  the 
center  of  the  arcs  AB  and  EF. 

Let  the  radius  GO  =  R  and  GO'  =  R',  then  BO  =  R  +  g. 


156  F11OGS    AND    SWITCHES. 

By  Article  139,  Fig.  92,  find  the  angles  F',  F",  and  F"f  of 
the  frogs  at  C,  D,  and  //  respectively  ;  also  the  chords  GC, 
GD,  CH,  and  the  side  OO',  and  angle  at  0'  of  the  triangle 
GOO'. 

In  the  triangle  GO' A, 

GA 


2GOf 
In  the  triangle  OO"O',  the  angle 

0'=OO'G+  GO' A, 
and  0"0'  —  0"0  —  E'  —  E  —  g  •  (167) 

for    O"0  =  /I  0'  —  A  0",    and    0"0  =  OB  —  0"B,    also    <9"£ 
=  A  O". 

Lay  off  0"P  =  0"0  on  CT0',  then 

0'F=R'-n  —  g.  (168) 

In  the  triangle  OPO', 

tan  J  (P  -  0)  =  00,  ~  Q/p  x  tan  ^90°  -  ~Y 

With  this  value  of  the  half  difference,  and  ^90° J  as 


the  value  of  the  half  sum,  the  angles  P  and  0  are  readily 
found.     Then 


The  exterior  angle  OPO"  =0+0'. 
The  isosceles  triangle  00"P  gives 

0"0=         OP 


2  cos  0"  OP 

Then  the  radius  0"B  of  the  arc  AB=OB—  00", 
or  0"£  =  R  +  c)  -  OO".  (169) 

The  radius  FO"  of  the  arc  EF=  0"B  —  </, 
or  FO"  =R-  OO".  (170) 

The  length  of  the  arc 

AB=  0"B  X  3.1416  X  4^'  (171) 

and  the  length  of  the  arc 

EF  =  0"F  X  .3.1416  X  4r'  (172) 


CROSSING   SLIPS. 


157 


Tf  the  tracks  cross  each  other,  as  in  Fig.  97,  the  solution 
will  be  the  same  as  above,  except  that  (167)  becomes 

O"(y  +  (JO  =  R  +  g  +  R',  (173) 

0"0'  =  AO'  +  A 0"  and  0"0  =  OB  +  0"B,  and  (168)  becomes 
OT^R  +  cj  +  R'.  (174) 

EXAMPLES. 

1.  Given,  in  Fig.  96,  a  2°  curved  track  crossed  by  a  2°  30' 
curved    track    making 

a  No.  8  frog  at  G, 
to  find  the  radii  and 
lengths  of  the  curved 
rails  AB  and  EF,  AG 
being  4'  10". 

2.  Given,  in  Fig.  97, 
a     1°      curved      track 
crossed    by    a     1°  30' 
curved    track    making 
a   No.  8   frog    at     G, 
to  find   the  radii  and 
lengths  of   the  curved 
rails  AB  and  EF,  AG 
being  5  feet. 


158 


TRIGONOMETRIC    FORMULAS. 


TRIGONOMETRIC    FORMULAS. 


FIG.  98. 


In  Fig  99,  let  DCE  be  the  arc  of  a  quadrant,  ABC  a  right 
triangle,  the  angle  BAC  subtended  by  the  arc  CE  =  A,  and 
consider  the  radius  A  C  =  unity.  Then 


AF=  cosecvl. 
BE  =  versin^l . 
DI  =coversin 
CH=exsecA. 

CF  = 


BC  =si 
AB  =  cos  A. 
HE  =  tan  A. 
DF  =  cot^l. 
AH  =  sec  A. 


Using  the  small  letters  a,  &,  c,  to  represent  the  sides  of  a 
right  triangle  in  Fig.  98  or  99,  we  may  write 


sin  A  =  -  •  cosecvl 
6 


c  b 

=  -;       secA=-; 


sin  A  = 


cos  A  — 


=  -;  .-.  tan^l  = 


cosec  A 

I 

secJ. 

I 
cot  4* 


SOLUTION    OF    TRIANGLES. 


159 


SOLUTION   OF  RIGHT  TRIANGLES. 


Required. 

Given. 

A,  C,c 

a,  6 

A,  C,  b 

a,  c 

C,  6,  c 
C,  a,  c 
C,  a,  6 

A,  a 
A,b 

A,c 

Formulas. 
smA=cosC  =  -;    c  =  «/(&+ a)  (6  —  a), 


tan  J.  =  cot  J5  =  - ;    b==  *a?  +  c'2. 
c 

(7  =  90°  — ^4;  c  = 
C  =  90°  —  A;  a= 
C  =  90°  — ^1  a  = 


Required. 

Given. 

6 

A,  B,  a 

* 

A,  a,  6 

K^-  -  ^) 

Ufc,CJ 

!       1 

J 

j       I 
i        r 

a,  6,  c 

Area 
Area 

A,  6,  c 

Area 

-4,  #,  c 

SOLUTION   OF  OBLIQUE  TRIANGLES. 

Formulas. 
asinJ5 


sin^l 

bsinA 


sin  B  = 


ta,ni(A  -  B)  = 


—  C) 

a  ~ 


a  +  b 
B)  +  \(A  —B} 
B)-i(A-  B) 


+  B) 


If 


c), 


tan-M  = 


=AF 

V" 


-V1 


sin  A  =  2x/s(s-a)(s-6)(.s-c) 
6c 


Area  =  *s  (s  —  a)  (s  —  b)(s  —  c) 
Area  =  |  be  sin  J. 

C'  sin  ^  sin  B 


Area  = 


2sin(     +.B) 


160  GENERAL   FORMULAS. 


GENERAL  FORMULAS. 


=-\l  —  cos:2  A  = 
sin  A  = 


sin  A  =  -  -  -  =  >/-Hl  —cos  A). 
cosec  A 


cos  A  =  —  =  —  =  •xfi  —  sin2  .4  = 
sec  A 


cos  A  =1—2  sin2-|-yl  =  1  —  versvl. 


cos  A  =  *£  4-  $cos2A  —  cos-jrA  —  sin2 


tan  ^4  =j    = 
cos  A 


cosyl  1  +  cos2A 


cot^l  sin  2^1 


t&nA       s'mA 

sin  2  ^4       =  1_ 
1  —  cos  2  A  sin  2  A 

sec  A  = =  the  reciprocal  of  any  expression  for  cos  A 

cos^l 

cosec  A  = =  the  reciprocal  of  any  expression  for  sin  ^4. 

sin  A 

\ersA  =  1  —  cos  A  =  2  sin2  \A. 

.. versA 

cos  A 


•V1 


sin|,4=.    »*--«»4 


GENERAL   FORMULAS.  161 


sin  A  1  —  cos  A 

sin  2^1  =  28*11^4  cos  A. 
cos  2  A  —  cos'2yl  —  sin2  A  =  2  cos2  A  —  1. 


1  —  tan2  A 
cot2  A  —  1 


2cot^l 

sin  ( A  ±  B)  =  sin  A  cos  B  ±  cos  A  sinB. 
cos  (A  ±  B)  =  cosA  cosB  q:  sin  A  sinB. 
tan  (A  ±  B)  =  tan  X±  tan  7*  § 


sin  J.  +  sin  B  =  2  sin  \  (A  +  7;?)  cos|  (yl  —  5). 

sin  ^4  -  sin  B  =  2cosl(A  +  7^)  sini(4  -  B). 

cos  A  +  co»B=2coslr(A  +  B)  cos±(A  —  B). 

cosB  —  cosA=2s'mi(A  +  B)  sm$(A  —  B). 

sin*  A  —  sin2  B  =  cos2  B  —  cos-  A  =  sin  (A  +  7?)  sin  (A  —  B), 

cos'2  A  —  s'm'2B  =  cos(A  +  B)  cos(A  —  B). 


cos  A  cosB 


sin  A  sinB 


162 


MISCELLANEOUS    FORMULAS. 


MISCELLANEOUS   FORMULAS. 


Required. 

Given. 

Formulas. 

Area  of 

Parallel  sides  =  m  and  n 

Trapezoid 

Perp.  dist.  bet.  them  =p 

-  (m  +  ?i) 

Regular  Polygon 

Length  of  side  =  I 
Number  of  sides  =  n 

nl'2     ,180° 
—  cot  
4           n 

Circle 

Radius  =  r 

Ttr2  [X  =  3.  1410 

Ellipse 

Semi-axes  =  a  and  6 

Ttab 

Parabola 

Base  =  b,  height  =  h 

\bh 

Surface  of 

Radius  of  base  =  r 

Cone 

Slant  height  =  s 

Ttrs 

Cylinder 

Radius  =  r,  height  =  h 

Z-rtrh 

Sphere 

Radius  =  r 

4rtr2 

Zone 

Height  =  h 

Zrtrh 

Radius  of  its  sphere  =  r 

Volume  of 
Prism  or  cylinder 

Area  of  base  =  b 
Height  =  h 

bh 

Pyramid  or  cone 

Area  of  base  =  b 
Height  =  h 

bh 
3 

Frustum  of 

Pyramid  or  cone 

Area  of  bases  =  b  and  &' 
Height  =  h 

3 

Sphere 

Radius  =  r 

irtr3 

TABLES. 


The  plates  for  Table  IX  and  for  I  and  II 
in  the  Appendix  were  prepared  by  Messrs. 
J.  S.  Gushing  &  Co.,  Norwood,  Mass.  All 
other  tables  except  Table  XI  are  printed 
from  electrotypes  furnished  by  Messrs. 
John  Wiley  and  Sons,  New  York. 


163 


TABLE  I.— RADII. 


Deg. 

ladius. 

Deg. 

ladius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

1°  0' 

nflnite 

0   Q/ 

5729.65 

0   ()/ 

2864.93 

3°  0' 

1910.08 

4°  0' 

1432.69 

U   1 

343775. 

1 

5635.72 

1 

2841.26 

1 

1899.53 

1 

1426.74 

2 

171887. 

2 

5544.83 

2 

2817.97 

2 

1889.09 

2 

1420.85 

3 

114592. 

3 

5456.82 

3 

2795.06 

3 

1878.77 

3 

1415.01 

4 

85943.7 

4 

5371.56 

4 

2772.53 

4 

1868.56 

4 

1409.21 

5 

68754.9 

5 

5288.92 

5 

2750.35 

5 

1858.47 

5 

1403.46 

6 

57295.8 

6 

5208.79 

6 

2728.52 

6 

1848.48 

6 

1397.76 

7 

49110.7 

7 

5131.05 

7 

2707.04 

7 

1838.59 

7 

1392.10 

8 

42971.8 

8 

5055.59 

8 

2685.89 

8 

1828.82 

8 

1386.49 

9 

38197.2 

9 

4982.33 

9 

2665.08 

9 

1819.14 

9 

1380.92 

10 

34377.5 

10 

4911.15 

10 

2644.58 

10 

1809.57 

10 

1375.40 

11 

31252.3 

11 

4841.98 

11 

2624.39 

11 

1800.10 

11 

1369.92 

12 

28647.8 

12 

4774.74 

12 

2604.51 

12 

1790.73 

12 

1364.49 

13 

26444.2 

13 

4709.33 

13 

2584.93 

13 

1781.45 

13 

1359.10 

14 

24555.4 

14 

4645.69 

14 

2565.65 

14 

1772.27 

14 

1353.75 

15 

22918.3 

15 

4583.75 

15 

2546.64 

15 

1763.18 

15 

1348.45 

16 

21485.9 

16 

4523.44 

16 

2527.92 

16 

1754.19 

16 

K343.15 

17 

20222.1 

17 

4464.  10 

17 

2509.47 

17 

1745.26 

17 

1337.65 

18 

19098.6 

18 

4407.46 

18 

2491.29 

18 

1736.48 

18 

1332.77 

19 

18093.4 

19 

4351.67 

19 

2473.37 

19 

1727.75 

19 

1327.63 

20 

17188.8 

20 

4297.28 

20 

2455.70 

20 

1719.12 

20 

1322.53 

21 

16370.2 

21 

4244.23 

21 

2438.29 

21 

1710.56 

21 

1317.46 

22 

15626.1 

22 

4192.47 

22 

2421.12 

22 

1702.10 

22 

1312.43 

23 

14946.7 

23 

4141.96 

23 

2404.19 

23 

1693.72 

23 

1307.45 

24 

14323.6 

24 

4092.66 

24 

2387.50 

24 

1685.42 

24 

1302.50 

25 

13751.0 

25 

4044.51 

25 

2371  .04 

25 

1677.20 

25 

1297.58 

26 

13222.1 

26 

3997.49 

26 

2354.80 

26 

1669.06 

26 

1292.71 

27 

12732.4 

27 

3951.54 

27 

2338.78 

27 

1661.00 

27 

1287.87 

28 

12277.7 

28 

3906.54 

28 

2322.98 

28 

1653.01 

28 

1283.07 

29 

11854.3 

29 

3862.74 

29 

2307.39 

29 

1645.11 

29 

1278.30 

30 

11459.2 

30 

3819.83 

30 

2292.01 

30 

1637.28 

30 

1273.57 

/31 

11089.6 

31 

3777.85 

31 

2276.84 

31 

1629.52 

31 

1268.87 

32 

1*0743  0 

32 

3736.79 

32 

2261.86 

32 

1621.84 

32 

1264.21 

33 

10417.5 

33 

3696.61 

33 

2247.08 

33 

1614.22 

33 

1259.58 

34 

10111.1 

34 

3657.29 

34 

2232.49 

34 

1606.68 

34 

1254.98 

35 

9822.18 

35 

3618.80 

35 

2218.09 

35 

1599.21 

35 

1250.42 

36 

9549.34 

36 

3581.10 

36 

2203.87 

36 

1591.81 

36 

1245.89 

37 

9291.29 

37 

3544.19 

37 

2189.84 

37 

1584.48 

37 

1241.40 

38 

9046.75 

38 

3508.02 

38 

2175.98 

38 

1577.21 

38 

1236.94 

39 

8814.78 

39 

3472.59 

39 

2162.30 

39 

1570.01 

39 

1232.51 

40 

8594.42 

40 

3437.87 

40 

2148.79 

40 

1562.88 

40 

1228.11 

41 

8384.80 

41 

3403.83 

41 

2135.44 

41 

1555.81 

41 

1223.74 

4-4 

8185.16 

42 

3370.46 

42 

2122.26 

42 

1548.80 

42 

1219.40 

43 

7994.81 

43 

3337.74 

43 

2109.24 

43 

1541.86 

43 

1215.30 

44 

7813.11 

44 

3305.65 

44 

2096.39 

44 

1534.98 

44 

1210.82 

45 

7639.49 

45 

3274.17 

45 

2083.68 

45 

1528.16 

45 

1206.57 

46 

7473.42 

46 

3243.29 

46 

2071.13 

46 

1521.40 

46 

1202.36 

47 

7314.41 

47 

3212.98 

47 

2058.73 

47 

1514.70 

47 

1198.17 

48 

7162.03 

48 

3183.23 

48 

20-16.48 

48 

1508.06 

48 

1194.01 

49 

7015.87 

49 

3154.03 

49 

20:M.37 

49 

1501.48 

49 

1189.88 

50 

6875.55 

50 

3125.36 

50 

2022.41 

50 

1494.95 

50 

1185.78 

51 

6740.74 

51 

3097.20 

51 

2010.59 

51 

1488.48 

51 

1181.71 

52 

6611.12 

52 

3069.55 

52 

1998.90 

52 

1482.07 

52 

1177.66 

53 

6486.38 

53 

3042.39 

53 

1987.35 

53 

1475.71 

53 

1173.65 

54 

6366.2 

54 

3015.71 

54 

1975.93 

54 

1469.41 

54 

1169.66 

55 

6250.5 

55 

2989.48 

55 

1964.64 

55 

1463.16 

55 

1165.70 

56 

6138.90 

56 

2963.71 

56 

1953.48 

»  56 

1456.96 

56 

1161.76 

57 

6031.2 

57 

2938.39 

57 

1942.44 

57 

1450.81 

57 

1157.85 

58 

5927.2 

58 

2913.49 

58 

1931.53 

58 

1444.72 

58 

1153.97 

59 

5826.7 

59 

2889.01 

59 

1920.75 

59 

1438.68 

K 

1150.11 

60 

5729.6 

60 

2864  -98 

60 

1910.08 

60 

1432.69 

60 

1146.28 

165 


TABLE   I.— RADII. 


Deg 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

5°0 

1146  28 

6°0 

955.37 

7°0 

818  G4 

8°0' 

716.34 

9°0' 

636.78 

I 

1142.47 

1 

952.72 

1 

816.70 

1 

714.85 

1 

635.61 

2 

1138  69 

2 

950.09 

2 

814.76 

2 

713.37 

2 

634.44 

3 

1134.94 

3 

947.48 

3 

812.83 

3 

711.90 

3 

633.27 

4 

1181.21 

4 

944.88 

4 

810.92 

4 

710.43 

4 

632.10 

5 

1127  50 

5 

942.29 

5 

809.01 

5 

708.96 

5 

630.94 

6 

1123.82 

6 

939.72 

6 

807.11 

6 

707,51 

6 

629.79 

7 

1120.16 

7 

937.16 

7 

805.22 

7 

706.05 

7 

6-28.64 

8 

1116.52 

8 

934.62 

8 

803.34 

8 

704.60 

8 

6:27.49 

9 

1112.91 

9 

932.09 

9 

801.47 

9 

703  16 

9 

626.35 

10 

1109.33 

10 

929.57 

10 

799.61 

10 

701.73 

10 

6:25.21 

11 

1105  76 

11 

927.07 

11 

797  75 

11 

700.30 

31 

624.08 

12 

1102.22 

12 

924.58 

12 

795.91 

1-2 

698.88 

12 

622.95 

13 

1098.70 

13 

922.10 

13 

794.07 

13 

697  46 

13 

621.82 

14 

1095.20 

-     14 

919.64 

14 

792.24 

14 

696.05 

14 

620.70 

15 

1091.73 

15 

917.19 

15 

790.42 

15 

694.65 

15 

619  58 

16 

1088.28 

16 

914.75 

16 

788.61 

16 

693.  -24 

16 

618.47 

17 

1084.85 

17 

912.33 

17 

786.80 

17 

691.85 

17 

617.36 

18 

1081.44 

18 

909.92 

18 

785.01 

18 

690.46 

18 

616  25 

19 

1078.05 

19 

907.52 

19 

783.22 

19 

689.08 

19 

615  15 

20 

1074.68 

20 

905.13 

20 

781.44 

20 

687.70 

20 

614.05 

21 

1071.34 

21 

902.76 

21 

779.67 

21 

686  33 

21 

612.96 

2:2 

1068.01 

22 

900.40 

22 

777  91 

22 

684  96 

22 

611  87 

23 

1064.71 

23 

898.05 

23 

776.15 

23 

683  60 

23 

610.78 

24 

1061  .43 

24 

895.71 

24 

774.40 

24 

682  25 

24 

609.70 

25 

1058.16 

25 

893.39 

25 

772.66 

25 

680.89 

25 

608.62 

26 

1054.92 

26 

891  .08 

26 

770.93 

26 

679.55 

26 

607.55 

27 

1051.70 

27 

888.78 

27 

769.21 

27 

678.21 

27 

606.48 

28 

1048.48 

28 

886.49 

28 

767.49 

28 

676.88 

28 

605.41 

29 

1045.311 

29 

884.21 

29 

765.78 

29 

675.54 

29 

604.35 

30 

1042.14 

30 

881.95 

30 

764.08 

30 

674  22 

30 

603.29 

31 

1039.00 

31 

879.69 

31 

760.39 

31 

672.90 

31 

602.23 

32 

1035.87 

32 

877.45 

32 

700.70 

32 

671.59 

32 

601.18 

33 

1032.76 

33 

875.22 

33 

759.02 

33 

670.28 

33 

600.13 

34 

1029.67 

34 

873.00 

34 

757.35 

34 

668.98 

34 

599.09 

35 

10sJ6  60 

35 

870.80 

35 

755.69 

35 

667.68 

35 

598.04 

36 

1023.55 

36 

868.60 

36 

754.03 

36 

666.39 

36 

597.01 

37 

1020.51 

37 

866.41 

37 

752.38 

37 

665.10 

37 

595.97 

38 

1017.49 

38 

864.24 

38 

750.74 

38 

603.82 

38 

594.94 

39 

1014.50 

39 

862.08 

39 

749.10 

39 

662.54 

.  39 

593.91 

40 

1011.51 

40 

859.92 

40 

747.48 

40 

661.S6 

40 

592.89 

41 

1008.55 

41 

857.78 

41 

745.86 

41 

659.99 

41 

591.87 

4-2 

1005.60 

4-2 

855.65 

42 

744.24 

42 

658.73 

42 

590.85 

43 

100-2.67 

43 

853.53 

43 

742.63 

43 

657.47 

43 

589.84 

44 

999.76 

44 

851.42 

44 

741.03 

44 

656.22 

44 

588.83 

45 

996.87 

45 

849.32 

45 

739.44 

45 

654.97 

45 

587.83 

46 

993.99 

46 

847.23 

46 

737.86 

46 

653.72 

46 

586.82 

47 

991.13 

47 

845.15 

47 

736.28 

47 

65-2.48 

47 

585.83 

48 

988.28 

48 

843.08 

48 

7:34.70 

48 

651.25 

48 

584.83 

49 

985.45 

49 

841.02 

49 

733.14 

49 

650.02 

49 

583.84 

50 

982.64 

50 

838.97 

50 

731.53 

50 

648.79 

50 

582.85 

51 

979.84 

51 

836.93 

51 

730.03 

51 

647.57 

51 

581.86 

52 

977.06 

52 

834.90 

52 

728.48 

52 

646.35 

52 

580.88 

53 

974.29 

53 

832.89 

53 

726.94 

53 

645.14 

53 

579.90 

54 

971.54 

54 

830  88 

54 

7-25.41 

54 

643.94 

54 

578.92 

55 

968.81 

55 

828.88 

55 

723.88 

55 

642.73 

55 

577.95 

56 

966.09 

56 

826.89  ' 

56 

72-2.36 

56 

641.53 

56 

576  98 

57 

963.39 

57 

824.91 

57 

720  85 

57 

640.34 

57 

576.02 

58 

960.70 

58 

822.93 

58 

719.34 

58 

639.15 

58 

575.06 

59 

958.03 

59 

8-.J0.97 

59 

717.84 

59 

637.96 

59 

574.10 

,    eo 

955.37 

60 

819.03 

60 

716.34 

60 

636.78 

60 

573.14 

166 


TABLE   I.— RADII. 


Dee. 

Radius.    Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Radius. 

Deg. 

Katiiur.. 

10°  0' 

573.14 

12°  0' 

477.68 

14°  0' 

409.32 

16°  0' 

358.17 

18°  0' 

818.31 

2 

571.24 

2 

476.36 

2 

408.35 

2 

357,43 

2 

317.80 

4 

5G9.35 

4 

475.05 

4 

407.38 

4 

356.69 

4 

317.2& 

6 

567  47 

6 

473.74 

6 

406.42 

6 

355.95 

6 

316.63 

8 

565  60 

8 

472.44 

8 

405.40 

8 

355.21 

8 

316  05 

10 

563.75 

10 

471.15 

10 

404.51 

10 

354.48 

10 

315.47 

12 

561  91 

12 

469  86 

12 

403.56 

12 

353.75 

12 

314.89 

14 

560.08 

14 

468.58 

14 

402.61 

14 

353  03 

14 

314  32 

16 

558.26 

16 

467.31 

16 

401.  G7 

16 

352.30 

16 

313  75 

18 

556.45 

18 

466.04 

18 

400.74 

18 

351.58 

18 

313.18 

20 

554.66 

20 

464.78 

20 

399.80 

20 

350.86 

20 

312.61 

22 

552.88 

22 

463.53 

22 

398.88 

22 

350.15 

22 

312.04 

24 

551.11 

24 

462  29 

24 

397.95 

24 

349.44 

24 

311.47 

26 

549.35 

26 

461.05 

'    26 

397.03 

26 

348.72 

26 

310  91 

28 

547.60 

28 

459.82 

28 

396.13 

28 

348.02 

28 

310  35 

30 

545.87 

30 

458.59 

30 

395.21 

30 

347.32 

30 

309.79 

32 

544.14 

32 

457.38 

32 

394.30 

32 

346.62 

32 

309.23 

34 

542.42 

34 

456.16 

34 

393.40 

34 

345.93 

34 

308.68 

36 

540.72 

36 

454.96 

36 

392.50 

36 

345.23 

36 

308/13 

38 

539.03 

38 

453.76 

38 

391.61 

38 

344.54 

38 

307.58 

40 

537.34 

40 

452.57 

40 

390.72 

40 

343.85 

40 

307.03 

42 

535.67 

42 

451.38 

42 

389.83 

42 

343.16 

42 

306.48 

44 

534.01 

44 

450.20 

44 

388.95 

44 

342.48 

44 

305.93 

46 

532.36 

46 

449.02 

46 

388.07 

46 

341.80 

46 

305.39 

48 

530.71 

48 

447.86 

48 

387.20 

48 

341.12 

48 

304.85 

50 

529.08 

50 

446  69 

50 

386.33 

50 

340.45 

50 

304.31 

52 

5:27.  46 

52 

445  54 

52 

385.47 

52 

339.78 

52 

303.77 

54 

525.85 

54 

444.39 

54 

384.60 

54 

339.11 

54 

303.24 

56 

524.25 

56 

443.54 

56 

383.75 

56 

338.44 

56 

302.70 

58 

522.65 

58 

442.11 

58 

882.89 

58 

337.77 

58 

302.17 

1100' 

521  .07 

13°  0' 

440.97 

16°  0' 

38-2.04 

17°0' 

337.11 

19°  0' 

301.64 

2 

519.50 

2 

439.85 

2 

381.19 

2 

336.45 

2 

301.13 

4 

517.93 

4 

438.73 

4 

380.35 

4 

335.80 

4 

300.59 

6 

516.38 

6 

437.61 

6 

379.51 

6 

335.14 

6 

300.07 

8 

514.84 

8 

436.50 

8 

378.68 

8 

334.49 

8 

299.  E4 

10 

513.30 

10 

435.40 

10 

377.84 

10 

333.84 

10 

299.0-2 

12 

511.77 

12 

434.30 

12 

377.02 

12 

333.19 

12 

298.50 

14 

510.26 

14 

433.21 

14 

376.19 

14 

332.55 

14 

297.99 

16 

503.75 

16 

432.12 

16 

375.37 

16 

331.91 

16 

297.47 

18 

507.25 

18 

431.04 

18 

374.55 

18 

331.  27 

18 

296.96 

20 

505.76 

20 

429.96 

20 

373.74 

20 

330.63 

20 

296.45 

22 

504.28 

22 

428.98 

22 

372.93 

22 

330.00 

22 

295.94 

24 

502.80 

24 

4-27.82 

24 

372.12 

24 

329.37 

24 

295.43 

26 

501.34 

26 

426.76 

26 

371.32 

26 

328.74 

26 

294.92 

28 

499.88 

28 

425.71 

28 

370.52 

28 

328.11 

28 

294.4:2 

30 

498.43 

30 

424.66 

30 

369.72 

30 

327.48 

30 

293.91 

32 

496.99 

3:2 

423.61 

32 

368.93 

32 

326.86 

32 

293.41 

34 

495.56 

34 

422.57 

34 

368.14 

34 

326.24 

34 

292.91 

36 

494.14 

36 

421.54 

36 

367.35 

36 

325.62 

36 

292.41 

38 

492.73 

38 

420.51 

38 

366.57 

38 

325.01 

38 

291.92 

40 

491.32 

40 

419.49 

40 

365.79 

40 

324.40 

40 

291.42 

42 

489.92 

42 

418.47 

42 

365.01 

42 

323.79 

42 

290.93 

44 

488.53 

44 

417.45 

44 

364.24 

44 

323.18 

44 

290.44 

46 

487.15 

46 

416.44 

46 

363.47 

46 

322.57 

46 

289.95 

48 

485.77 

48 

415.44 

48 

362.70 

48 

321.97 

48 

289.46 

50 

484.40 

50 

414.44 

50 

361.94 

50 

321.37 

50 

288.98 

tt) 

483.05 

52 

413.44 

52 

361.18 

52 

320.77 

52 

288.49 

54 

481.69 

54 

412.45 

54 

360.42 

54 

320.17 

54 

288.01 

56 

480.35 

56 

411.47 

56 

359.67 

56 

319.57 

56 

287.53 

58 

479.01 

58 

410.49 

58 

358.92 

58 

318.98 

58 

287.05 

60 

477.68 

60 

409.51 

60 

358.17 

60 

318.39 

60 

286.57 

I 

TABLE  II.— TANGENTS  AND  EXTERNALS  TO  A  1°  CURVE. 


Angle. 

Tan- 
gent. 

Exter- 
nal. 

Angle. 

Tan- 
gent. 

Exter- 
nal. 

Angle. 

Tan- 
gent. 

Exter- 
nal. 

oc 

T. 

E. 

oc 

T. 

E. 

oc 

T. 

E. 

1° 

50.00 

.218 

11 

551.70 

26.500 

21° 

1061.9 

97.577 

10' 

58.34 

.297 

10' 

560.11 

27.313 

!(/ 

1070.6 

99.155 

20 

66.67 

.388 

20 

568.53 

28.137 

-  20 

1079.2 

100.75 

30 

75.01 

.491 

30 

576.95 

28.974 

30 

1087.8 

102.35 

40 

as.  34 

.606 

40 

585.36 

29.824 

40 

1096.4 

103.97 

50 

91.68 

.733 

50 

593.79 

30.686 

50 

1105.1 

105.60 

2 

100.01 

.873 

12 

602.21 

31.561 

22 

1113.7 

107.24 

10 

108.35 

1.024 

10 

610.64 

32.447 

10 

1122.4 

108.90 

20 

116.68 

1.188 

20 

619.07 

33.347 

20 

1131.0 

110.57 

30 

125.02 

1.364 

30 

627.50 

34.259 

30 

1139.7  !  112.25 

40 

133.36 

1.552 

40 

635.93  '35.183 

40 

1148.4     113.95 

50 

141.70 

1.752 

50 

644.37 

36.120 

50 

1157.0     115.66 

3 

150.04 

1.964 

13 

652.81 

37.070 

23 

1165.7 

117.38 

10 

158.38 

2.188 

10 

661.25     38.031 

10 

1174.4 

119.12 

20 

166.72 

2.425 

20 

669.70     39.006 

20 

1183.1 

120.87 

30 

175.06 

2.674 

30 

678.15 

39.993 

30 

1191.8 

122.63 

40 

183.40 

2.934 

40 

686.6.) 

40.992 

40 

1200.5 

124.41 

50 

191.74 

3.207 

50 

695.06 

42.004 

50 

1209.2 

12C.20 

4 

200.08 

3.492 

14 

703.51 

43.029 

24 

1217.9 

128.00 

10 

208.43 

3.790 

10 

711.97 

44.066 

10 

1226.6 

129.82 

20 

216.77 

4.099 

20 

720.44 

45.116 

20 

1235.3 

131.65 

30 

225.12 

4.421 

30 

728.90 

46.178 

30 

1244.0  1  133.50 

40 

233.47 

4.755 

40 

737.37 

47.253 

40 

1252.8     135.35 

50 

241.81 

5.100 

50 

745.85 

48.341 

50 

1261.5 

137.23 

5 

250.16 

5.459 

15 

754.32 

49.441 

25 

1270.2 

139.11 

10 

258.51 

5.829 

10 

762.80 

50.554 

10 

1279.0 

141.01 

20 

266.86 

6.211 

20 

771.99 

51.679 

20 

1287.7 

142.93 

30 

275.21 

6.606 

30 

779.77 

52.818 

30 

1296.5 

144.85 

40 

283.57 

7.013 

40 

788.26 

53.969 

40 

1305.3 

146.79 

50 

291.92 

7.432 

50 

796.75 

55.132 

50 

1314.0 

148.75 

6 

300.28 

7.863 

16 

805.25 

56.309 

26 

1322.8 

150.71 

10 

308.64 

8.307 

10 

813.75     57.498 

10 

1331.6 

152.69 

20 

316.99 

8.762 

20 

822.25     58.699 

20 

1340.4 

154.69 

30 

325.35 

9.230 

30 

830.76     59.914 

30 

1349.2 

156.70 

40 

333.71 

9.710 

40 

839.27 

61.141 

40 

1358.0 

158.72 

50 

342.08 

10.202 

50 

847  ..78 

62.381 

50 

1366.8 

160.76 

7 

350.44 

10.707 

17 

856.30 

63.634 

27 

1375.6 

162.81 

10 

358.81 

11.224 

10 

864.82  i  64.900 

10 

1384.4 

164.86 

20 

367.17 

11.753 

20 

873.35  |  66.178 

20 

1393.2 

166.95 

30 

375.54 

12.294 

30 

881.88     67.470 

30 

1402.0 

169.04 

40 

383.91 

12.847 

40 

890.41 

68.774 

40 

1410.9 

171.15 

50 

392.28 

13.413 

50 

898.95 

70.091 

50 

1419.7 

173.27 

8 

400.66 

13.991 

18 

907.49 

71.421 

28 

14:28.6 

175.41 

10 

409.03 

14.582 

10 

916.03 

72.764 

10 

1437.4 

177  .  55 

20 

417.41 

15.184 

20 

924.58 

74.119 

20 

1446.3 

179.72 

30 

425.79 

15.799 

30 

933.13 

75.488 

30 

1455.1 

181.89 

40 

434.17 

16.426 

40 

941.69 

76.869 

40 

1484.0 

184.08 

50 

442.55 

17.065 

50 

950.25 

78.264 

50 

1472.9 

186.29 

9 

450.93 

17.717 

19 

958.81 

79.671 

29 

1481.8 

188.51 

10 

459.32 

18.381 

10 

967.38 

81.092 

10 

1490.7 

190.74 

20 

467.71 

19.058 

20 

975.96 

82.525 

20 

1499.6 

192.99 

30 

476.10 

19.746 

30 

984.53 

83.972 

30 

1508.5 

195.25 

40 

484.49 

20.447 

40 

993.12 

85.431 

40 

1517.4 

197.53 

50 

492.88 

21.161 

50 

1001.7 

86.904 

50 

1526.3 

199.82 

10 

501.28 

21.887 

20 

1010.3 

88.389 

30 

1535.3 

202.12 

10 

509.68 

22.624 

10 

1018.9 

89.888 

10 

1544.2 

204.44 

20 

518.08 

23.375 

20 

1027.5 

91.399 

20 

1553.1 

206.77 

30 

526.48 

24.138 

30 

1036.1 

92.924 

30 

1562.1 

209.12 

40 

534.89 

24.913 

40 

1044.7 

94.462 

40 

1571.0 

211.48 

50 

543.29 

25.700 

50 

1053.3 

96.013 

50 

1580.0 

213.80 

168 


TABLE  II.— TANGENTS  AND  EXTERNALS  TO  A  1°  CURVE. 


Angle. 
<x 

Tan- 
gent. 

T. 

Exter- 
nal. 

E. 

Angle. 

oc 

Tan- 
gent. 

T. 

Exter- 
nal. 

E. 

Angle, 
oc 

Tan- 
gent. 

T. 

1 

Exter- 
nal. 

E. 

31° 

1589.0 

216.25 

41° 

2142.2 

387.38 

51° 

2732.9 

618.39 

10 

1598.0 

218.66 

10' 

2151.7 

390.71 

10' 

2743.1 

622.81 

20 

1606.9 

221.08 

20 

2161.2 

394.06 

20 

2753.4 

627.24 

30 

1615.9 

223.51 

30 

2170.8 

397.43 

30 

2763.7 

631.69 

40 

1624.9 

225.96 

40 

2180.3 

400.82 

40 

2773.9 

636.17 

50 

1633.9 

228.42 

50 

2189.9 

404.22 

-    50 

2784.2 

640.66 

82 

1643.0 

230.90 

42 

2199.4 

407.64 

52 

2794.5 

645.17 

10 

1652.0 

233.39 

10 

2209.0 

411.07 

10 

2804.9 

649.70 

20 

1661.0 

235.90 

20 

2218.6 

414.52 

20 

2815.2 

654.25 

30 

16TO.O 

238.43 

30 

2228.1 

417.99 

30 

2825.6 

658.83 

40 

1679.1 

240.96 

,40 

2237.7 

421.48 

40 

2835.9 

663.42 

50 

1688.1 

243.52 

(60 

2247.3 

424.98 

50 

2846.3 

668.03 

83 

1697.2 

246.08 

43 

2257.0 

428.50 

53 

2856.7 

672.66 

10 

1706.3 

248.66 

10 

2266.6 

432.04 

10 

2867.1 

677.32 

20 

1715.3 

251.26 

20 

2276.2 

435.59 

20 

2877.5 

681.99 

30 

1724.4 

253.87 

30 

2285.9 

439.16 

30 

2888.0 

686.68 

40 

1733.5 

256.50 

40 

2295.6 

422.75 

40 

2898.4 

691.40 

50 

1742.6 

259.14 

50 

2305.2 

446.35 

50 

2908.9 

693.13 

84 

1751.7 

261.80 

44 

2314.9 

449.98 

54 

2919.4 

700.89 

10 

1760.8 

264.47 

10 

2324.6 

453.62 

10 

2929.9 

705.66 

20 

1770.0 

267.16 

20 

2334.3 

457.27 

20 

2940.4 

710.46 

30 

1779.1 

269.86 

30 

2344.1 

460.95 

30 

2951.0 

715.28 

40 

1788.2 

272.58 

40 

2353.8 

464.64 

.      40 

2961.5 

720.11 

50 

1797.4 

275.31 

50 

2363.5 

468.35 

50 

2972.1 

724.97 

35 

1806.6 

278.05 

45 

2373.3 

472.08 

55 

2982.7 

729.85 

10 

1815.7 

280.82 

10 

2383.1 

475.82 

10 

2993.3 

7'34.76 

20 

1824.9 

283.60 

20 

2392.8 

479.59 

20 

3003.9 

739.68 

30 

1834.1     286.39 

30 

2402.6 

483.37 

30 

3014.5 

744.62 

40 

1843.3  I  289.20 

40 

2412.4 

487.17 

40 

3025.2 

749.59 

50 

1852.5 

292.02 

50 

2422.3 

490.98 

50 

3035.8 

754.57 

36 

1861.7 

294.86 

46 

2432.1 

494.82 

56 

3046.5 

759.58 

10 

1870.9 

297.72 

10 

2441.9 

498.67 

10 

3057.2 

764.61 

20 

1880.1 

300.59 

20 

2451.8 

502.54 

20 

3067.9 

769.66 

30 

1889.4 

303.47 

30 

2461.7 

506.42 

30 

3078.7 

774.73 

40 

1898.6 

306.37 

40 

2471.5 

510.33 

40 

3089.4 

779.83 

50 

1907.9 

309.29 

50 

2481.4 

514.25 

50 

3100.2 

784.94 

87 

1917.1 

312.22 

47 

2491.3     518.20 

57 

3110.9 

790.08 

10 

1926.4 

315.17 

10 

2501.2 

522.16 

10 

3121.7 

795.24 

20 

1935.7 

318.13 

20 

2511.2 

526.13 

20 

3132.6 

800.42 

30 

1945.0 

321.11 

30 

2521.1 

530.13 

30 

3143.4 

805.62 

40 

1954.3 

324.11 

40 

2531.1 

534.15 

40 

3154.2 

810.85 

50 

1963.6 

327.12 

50 

2541.0 

538.18 

50 

3165.1 

816.10 

88 

1972.9 

330.15 

43 

8561.0 

542.23 

58 

3176.0 

821.37 

10 

1982.2 

333.19 

10 

2561.0 

546.30 

10 

8186.  9 

826.66 

20 

1991.5 

a36.25 

20 

2571.0 

550.39 

20 

3197.8 

831.98 

30 

2000.9 

339.32 

30 

2581.0 

554.50 

30 

3208.8 

837.31 

40 

2010.2 

342.41 

40 

2591.1 

558.63 

40 

3219.7 

842.67 

50 

2019.6 

345.52 

50 

2601.1 

562.77 

50 

3230.7 

848.06 

39 

2029.0 

348.64 

49 

2611.2 

566.94 

59 

3241.7 

853.46 

10 

2038.4 

351.78 

10     2021.2 

571.12 

10 

3252.7 

858.89 

20 

2047.8 

354.94 

20     2631.3 

575.32 

20 

3263.7 

864.34 

30 

2057.2 

358.11 

30     2641.4 

579.54 

30 

3274.8 

869.82 

40 

2066.6     361.29 

40     2651.5 

583.78  I 

40 

3285.8 

875.32 

50 

2076.0 

3&4.50 

50 

2661.6 

588.04 

50 

3296.9 

880.84 

40 

2085.4 

367.72 

50 

2671.8 

592.32  i 

60 

3308.0 

886.38 

10 

2094.9 

370.95 

10 

2681.9 

596.62  j 

10 

3319.1 

891.95 

20 

2104.3     374.20 

20 

2692.1 

600.93  j 

20 

3330.3 

897.64 

30 

2113.8     377.47 

30 

2702.3 

605.27  1 

30 

3341.4 

903.15 

40 

2123.3 

380.76 

40 

2712.5 

609.62  i 

40 

3352.6 

908.79 

50 

2132.7 

384.06 

50 

2722.7 

614.00  ! 

50 

8363.8 

914.45 

TABLE  II.— TANGENTS  AND  EXTERNALS  TO  A  1°  CURVE. 


Angle. 

oc 

Tan- 
gent. 

T. 

Exter- 
nal. 

E. 

Angle, 
cc 

Tan- 
gent. 

T. 

Exter- 
nal. 

E. 

Angle. 

oc 

Tan- 
gent. 

T. 

Exter- 
nal. 

E. 

61° 

3375.0 

920.14 

71° 

4086.9 

1308.2 

81° 

4893.6 

1805.3 

10' 

3386.3 

925.85 

10' 

4099.5 

1315.6 

10' 

4908.0 

1814.7 

20 

3397.5 

931.58 

20 

4112.1 

1322.9 

20 

4922.5 

1824.1 

30 

'3408.8 

937.34 

30 

4124.8 

1330.3 

30 

4937.0 

1833  6 

40 

3420.1 

943.12 

40 

4137.4 

1337.7 

40 

4951.5 

1843.1 

50 

3431.4 

948.92 

50 

4150.1 

1345.1 

50 

4966.1 

1852.6 

62 

3442.7 

954.75 

72 

4162.8 

1352.6 

82 

4980.7 

1862.2 

10 

3454.1 

960.60 

10 

4175.6 

1360.1 

10 

4995.4 

1871.8 

20 

3465.4 

966.48 

20 

4188.5 

1367.6 

20 

5010.0 

1881.5 

30 

3476.8 

972.38 

30 

4201.2 

1375.2 

30 

5024.8 

1891.2 

40 

3488.3 

978.31 

40 

4214.0 

1382.8 

40 

5039.5 

1900.9 

50 

3499.7 

984.27 

50 

4226.8 

1390.4 

50 

5054.3 

1910.7 

63 

3511.1 

990.24 

73 

4239.7 

1398.0 

83 

5069.2 

1920.5 

10 

3522.6 

996.24 

10 

4252.6 

1405.7 

10 

5084.0 

1930.4 

20 

3534.1 

1002.3 

20 

4265.6 

1413.5 

20 

5099.0      1940.3 

30 

3545.6 

1008.3 

30 

4278.5 

1421.2 

30 

5113.9      1950.3 

40 

3557.2 

1014.4 

40 

4291.5 

1429.0 

40  |  5128.9 

1960.2 

50 

3568.7 

1020.5 

50 

4304.6 

1436.8 

50  i  5143.9 

1970.3 

64 

3580.3 

1026.6 

74 

4317.6 

1444.6 

84         1  5159.0 

1980.4 

10 

3591.9 

1032.8 

10 

4330.7 

1452.5 

10     5174.1 

1990.5 

20 

3603.5 

1039.0 

20 

4343.8 

1460.4 

20     5189.3 

2000.6 

30 

3615.1 

1045.2 

30 

4356.9 

1468.4 

30 

5204.4 

2010.8 

40 

3626.8 

1051.4 

40 

4370.1 

1476.4 

40 

5219.7 

2021.1 

50 

3638.5 

1057.7 

50 

4383.3 

1484.4 

50 

5234.9 

2031.4 

65 

3650.2 

1063.9 

75 

4396.5 

1492.4 

85 

5250.3  !  2041.7 

10 

3661.9 

1070  2 

10 

4409.8 

1500.5 

10 

5265.6     2052.1 

20 

3673.7 

1076.6 

20 

4423.1 

1508.6 

20  1  5281.0     2062.5 

30 

3685.4 

1082.9 

30 

4436.4 

1516.7 

30  1  5296.4     2073.0 

40 

3697.2 

1089.3 

40 

4449.7 

1524.9 

40  i  5311.9     2083  5 

50 

3709.0 

1095.7 

50 

4463.1 

1533.1 

50      5327.4 

2094.1 

66 

3720.9 

1102.2 

76 

4476.5 

1541.4 

86 

5343.0 

2104.7 

10 

3732.7 

1108.6 

10 

4489.9 

1549.7 

10 

5358.6 

2115.3 

20 

3744.6 

1115.1 

20 

4503.4 

1558.0 

20      5374.2 

2126.0 

30 

3756.5 

1121.7 

30 

4516.9 

1566.3 

30 

5389.9 

2136.7 

40 

3768.5 

1128.2 

40 

4530.4 

1574.7 

40 

5405.6 

2147.5 

50 

3780.4 

1134.8 

50 

4544.0 

1583.1 

50 

5421.4 

2158.4 

67 

3792.4 

1141.4 

77 

4557.6 

1591.6 

87 

5437.2 

2169.2 

,  10 

3804.4 

1148.0 

10 

4571.2 

1600.1 

10 

5453.1 

2180.2 

20 

3816.4 

1154.7 

20 

4584.8 

1608.6 

20 

5469.0     2191.1 

30 

3828.4 

1161.3 

30 

4598.5 

1617.1 

30 

5484.9 

2202.2 

40 

3840.5 

1168.1 

40 

4612.2 

1625.7 

40 

5500.9 

2213.2 

50 

3852.6 

1174.8 

50 

4626.0 

1634.4 

50 

5517.0 

2224.3 

68 

3864.7 

1181.6 

78 

4639.8 

1643.0 

88 

5533.1 

2235.5 

10 

3875.8 

1188.4 

10 

4653.6 

1651.7 

10 

5549.2 

2246.7 

20 

3889.0 

1195.2 

20 

4667.4 

1660.5 

20 

5565.4 

2258.0 

30 

3901.2 

1202.0 

30 

4681.3 

1669.2 

30 

5581.6 

2269.3 

40 

3913.4 

1208.9 

40 

4695.2 

1678.1 

40 

5597.8 

2280.6 

50 

3925.6 

1215.8 

50 

4709.2 

1686.9 

50 

5614.2 

2292.0 

39 

3937.9 

1222.7 

79 

4723.2 

1695.8 

89 

5630.5 

2303  5 

10 

3950.2 

1229.7 

10 

4737.2 

1704.7 

10 

5646.9 

2315.0 

20 

3962.5 

1236.7 

20 

4751.2 

1713.7 

20 

5663.4 

2326.6 

30 

3974.8 

1243.7 

30 

4765.3 

1722.7 

30 

5679.9 

2338.2 

40 

3987.2 

1250.8 

40 

4779.4 

1731.7 

40 

5696  4 

2349.8 

50 

3999.5 

1257.9 

50 

4793.6 

1740.8 

50 

5713.0 

2361.5 

70 

4011.9 

1265.0 

80 

4807.7 

1749.9 

90 

5729.7 

2373.3 

10 

4024.4 

1272.1 

10 

4822.0 

1759.0 

10 

5746.3 

2385.1 

20 

4036.8 

1279.3 

£0 

4836.2 

1768.2 

20 

5763.1 

2397.0 

30 

4049.3 

1286.5 

30 

4850.5 

1777.4 

30 

5779.9 

2408.9 

40 

4061.8 

1293.6 

40 

4864.8 

1786.7 

40 

5796.7 

2420.9 

50 

4074.4 

1300.9 

50 

4879.2 

1796.0 

50 

5813.6 

2432.9 

170 


'TABLE  H.— TANGENTS  AND  EXTERNALS  TO  A  1°   CURVE. 


Angle. 

Tan- 
gent. 

Ex- 
ternal. 

Angle. 

Tan- 
gent. 

Ex- 
ternal. 

Angle. 

Tan- 
gent. 

Ex- 
ternal 

cc 

T. 

E. 

oc 

T. 

E. 

oc 

T. 

E. 

91° 

5830.5 

2444 

9 

97 

6476.2 

2917.3 

103 

7203. 

2 

34" 

'4.4 

10' 
20 

5847.5 
5864.6 

2457.1 

2469.3 

10 
20 

6495.2 
6514.3 

2931.6 
2945.9 

10 
20 

7224.7 
7246.3 

3491.3 
3508.2  * 

30 

5881.7 

2481 

.5 

30 

65? 

J3.4 

2960.3 

30 

7268. 

0 

3525.2 

i 

0 

5£ 

98.8 

2493 

8 

i 

) 

65, 

52.6 

2974. 

r 

40 

7289. 

8 

35^ 

12.4 

50 

5916.0 

2506.1 

50 

65 

"1.9 

2989.2 

50 

7311. 

7 

3559.6 

92 

10 
20 

5933.2 
5950.5 
5967.9 

2518 
2531 
2543 

5 
.0 
.5 

98 

10 
20 

6591.2 
6610.6 
6630.1 

3003  8 
3018.4 
3033.1 

104 

10 
20 

7333.6 
7355.6 
7377.8 

3576.8 
3594.2 
3611.7 

30 

5985.3 

2556 

.0 

30 

6649.6 

3047. 

3 

30 

7399 

9 

36 

29.2 

4 

0 

6( 

)02.7 

2568 

.6 

4! 

3 

66 

39.2 

3062. 

3 

40 

7422 

2 

86 

46.8 

50 

6020.2 

2581 

.8 

50 

6688.8 

3077.7 

50 

7444 

6 

3664.5 

93 

10 

6037.8 
6055.4 

2594.0 
2606.8 

99 

10 

6708.6 
6728.4 

3092.7 
3107.7 

108  10 

7467.0 
7489.6 

3682.3 
3700.2 

1 

JO 

1 

)73.1 

2619 

.7 

2 

0 

67 

48.2 

3122. 

9 

20 

7512 

2 

37 

ia.2 

; 

JO 

6C 

)90.8 

2G32 

.6 

3 

a 

67 

38.1 

3138. 

1 

30 

7534 

9 

37 

36.2 

40 

6108.6 

2645 

.5  ! 

40 

6788.1 

3153.3 

40 

7557 

7 

3754.4 

)0 

6 

L26.4 

2058 

.5 

5 

0 

68 

38.2 

3168. 

7 

50 

7580 

5 

sr 

72.6 

94 

6 

144.3 

2671 

.6 

100 

68 

28.3 

3184. 

1 

1 

96 

7603 

5 

37 

91.0 

10 

6162.2 

2684 

.7 

10 

6848.5 

3199.6 

10 

7626.6 

3809.4 

1 

JO 

6 

180.2 

2697 

.9 

2 

0 

68 

S8.8 

3215. 

1 

20 

7649 

7 

38 

27.9 

30 

6198.3 

2711 

.2 

3 

0 

6889.2 

3230.8 

30 

7672.9 

a 

46.5 

to 

6 

216.4 

2724 

.5 

4 

0 

69< 

(39.6 

3246. 

5 

40 

7696 

.3 

3i* 

65.2 

50 

6234.6 

2737 

.9 

50 

69 

30.1 

3262. 

3 

50 

7719 

.7 

3884.0 

95 

6252.8 

2751 

.3 

101° 

6950.6 

3278. 

1 

107 

7743 

.2 

3902.9 

10 

6 

271.1 

2764 

.8 

1 

0' 

69 

71.3 

3294. 

1 

10 

7766 

.8 

3£ 

21.9 

20 

6289.4 

2778.3 

20 

6992.0 

3310. 

20 

7790 

.5 

3940.9 

30 

6. 

307.9 

2792 

.0 

3 

0 

70 

12.7 

3326. 

1 

30 

7814 

.3 

3£ 

60.1 

40 

6 

326.3 

2805 

.6 

4 

0 

n 

33.6 

3342. 

3 

40 

7838 

.1 

at 

79.4 

50 

& 

344.8 

2819 

.4 

1 

0 

70 

54.5 

3358. 

5 

50 

78G2 

.1 

31 

>98.7 

96 

6363.4 

2833.2 

102 

7075.5 

3374. 

9 

108 

7886 

.2 

4018.2 

10 

6, 

382.1 

2847 

.0 

1 

0 

70 

96.6 

3391. 

2 

10 

7910 

.4 

4C 

)37.8 

20 

6400.8 

2861.0 

20 

7117.8 

3407. 

7 

20 

7934 

.6 

4057.4 

30 

6 

419.5 

287E 

.0 

$ 

0 

71 

39.0 

3424. 

3 

30 

7959 

.0 

4C 

yr7.2 

40 

6 

438.4 

288£ 

4 

0 

71 

60.3 

3440. 

9 

40 

7983 

.5 

4( 

KW.l 

50 

6457.3 

290J 

.1 

50 

7181.7 

3457. 

6 

50 

8008.0 

4117.0 

CORRECTIONS  FOR  TANGENTS  AND  EXTERNALS. 

FOB  TANGENTS,  ADD 

FOR 

EXTERNALS, 

ADD 

Ang 

5° 

10° 

15° 

20° 

25° 

30° 

Ang 

5° 

10° 

15° 

20° 

25° 

30° 

oc 

Cur. 

Cur. 

Cur. 

Cur.  Cur. 

Cur. 

GC 

Cur. 

Cur. 

Cur. 

Cur. 

Cur. 

Cur. 

10° 

.03 

.06 

09~ 

.1 

3      .16 

.19 

10° 

001 

.003 

.004 

.006 

.007 

.008 

20 

.06 

.13 

'.19 

.2 

3      .32 

.39 

20 

.006 

.011 

.017 

.022 

.028 

.034 

30 

.10 

.19 

.29 

.3 

3      .49 

.59 

30 

.013 

.025 

.038 

.051 

.065 

.078 

40 

.13 

.26 

.40 

.5. 

3      .67 

80 

1  40 

.023 

.046 

.070 

.093 

.117 

.141 

50 

.17 

.34 

.51 

.6 

3      .85 

1  02 

j  50 

.037 

.075 

.116 

.151 

.189 

.227 

60 

.21 

.42 

.63 

.8 

4    1.05 

1.27 

60 

.056 

.112 

.168 

.225 

.283 

.340 

70 

.25 

.51 

.76 

1.0 

2    1.28 

1.54 

i  70 

.080 

.159 

.240 

.321 

.403 

,.485 

80 

.30 

.61 

.91 

1.22    1.53 

1.84 

80 

.110 

.220 

.332 

.445 

.558 

.671 

90 

.36 

.72 

1.09 

1.4 

5    1.83 

2.20 

90 

.149 

.299 

.450     .603 

.756 

.910 

100 

.43 

.86 

1.30 

1.7 

4   2  18 

2.62 

100 

.200 

.401 

.604     .809 

1.015 

1.221 

110 

.51 

1.03     1.56 

2  0 

8   2.61 

3.14 

110 

.268 

.536 

.806  1.082 

1.355 

1.633 

120 

.62 

1.25 

1.93 

2.5 

2   3  16 

3.81 

120 

,360 

.721 

1.086  1.456 

1.825 

2.197 

TABLE  III.  ^TANGENTIAL  OFFSETS  100  FT.  ALONG  THE  CURVE. 


Deg.  of 
Curve. 

0' 

10' 

20' 

30' 

40' 

50' 

0° 

0 

000 

0.145 

0.291 

0< 

136 

0.582 

0.727 

1° 

0 

873 

1.01 

8 

1.164 

i!i 

J09 

1.454 

1 

.600 

2° 

1 

745 

1.891 

2.036 

2.181 

2.327 

2.472 

3° 

2 

618 

2.76 

I 

2.908 

3  ( 

154 

3.199 

a 

.345 

40 

3 

490 

3.63 

5 

3.781 

3. 

4.071 

4 

.217 

5° 

4 

362 

4.507 

4  ."653 

4.798 

4.943 

5.088 

6° 

5 

234 

5.37 

D 

5.524 

5. 

369 

5.814 

E 

.960 

r-o 

6 

105 

6.250 

6.395 

6.540 

6.685 

6 

.831 

8° 

6 

976 

7.1* 

1 

7.206 

7 

111 

7.556 

.701 

9° 

7 

846 

7.99 

1 

8.136 

8. 

281 

8.426 

I 

.57i 

10° 

8 

716 

8.860 

9.005 

9.150 

9.295 

1 

.440 

11° 

9 

585 

9.72 

9 

9.874 

10. 

)19 

10.164 

1C 

.308 

12° 

10 

453 

10.59 

1 

0.742 

10. 

387 

11.031 

11 

176 

13° 

11 

320 

11.465 

11.609 

11.754 

11.898 

12.043 

14° 

12 

187 

12.33 

1 

1 

2.476 

12. 

120 

12.764 

12 

.908 

15° 

13 

053 

13.197 

13.341 

13.485 

13.629 

13.773 

16° 

13 

917 

14.06 

1 

1 

4  205 

14. 

349 

14.493 

14 

.637 

17° 

14 

781 

14.92 

5 

1 

5.069 

15. 

212 

15.356 

15 

.500 

18° 

15.643 

15.787 

15.931 

16.074 

16.218 

16.301 

19° 

16 

505 

16.64 

8 

16.792 

16.935 

17.078 

17 

.224 

20° 

17 

365 

17.50 

8 

1 

7.651 

17. 

'94 

17.937 

if 

.081 

21° 

18 

224 

18.36 

1 

8.509 

18. 

352 

18.795 

1£ 

.938 

22° 

19 

081 

19.224 

19.366 

19.509 

19.652 

19.794 

23° 

19 

937 

20.07 

i) 

I 

0.222 

20. 

364 

20.507 

20 

.649 

24° 

20 

791 

20.933 

21.076 

21.218 

21.360 

21 

.502 

TABLE  IV.—  MID-ORDINATES  TO  A  100-FT.   CHORD. 

Dof 

0 

1 

2 

3 

4 

5 

6           7 

8' 

9 

Curve. 

0° 

0.000 

0.21* 

0.436 

0.655 

0.873 

1  091 

1.309    1.528 

1.746 

1.965 

10° 

2.183 

2.40x 

2.620 

2.839 

3.058 

3.277 

3.496    3.716 

3.935 

4.155 

20° 

4.374 

4,594 

4.814 

5.035 

5.255 

5.476 

5.697    5.918 

6.139 

6.360 

Note.— As  an  example  illustrating  the  use  of  Table  IT,  suppose  we 
require  the  value  of  T  for  a  5°  curve,  where  /  =  40°  20'.    Then 


2104.3 


+  .13  =  420.99. 


TABLE  V.-LONG  CHORDS. 


Degree 
of 
Curve. 

Actual 
Arc, 
One 
Station. 

LONG  CHORDS. 

2 

Stations. 

3 

Stations. 

4 

Stations. 

5 

Stations. 

6 

Stations. 

Q°W 

100.000 

200.000 

299.999 

399.998 

499.996 

599.993 

20 

.000 

199.999 

299.997 

399.992 

499.983 

599.970 

30 

.000 

199.998 

299.992 

399.981 

499.962 

599.933 

40 

.001 

199.997 

299.986 

399.966 

499.932 

599.882 

50 

.001 

199.995 

299.979 

399.947 

499.894 

599.815 

1 

100.001 

199.992 

299.970 

399.924 

499.848 

599.733 

10 

.002 

199.990 

299.959 

399.896 

499.793 

599.637 

20 

.002 

199.986 

299.946 

399.865 

499.729 

599.526 

SO 

.003 

199.983 

299.932 

399.829 

499.657 

599.401 

40 

.003 

199.979 

299.915 

399.789 

499.577 

599.260 

50 

.004 

199.974 

299.898 

399.744 

499.488 

599.105 

2 

100.005 

199.970 

299.878 

399.695 

499.391 

598.934 

10 

.006 

199.964 

299.857 

399.643 

499.285 

598.750 

20 

.007 

199.959 

299.83-1 

399.586 

499.171 

598.550 

30 

.008 

199.952 

299.810 

399.524 

499.049 

598.336 

40 

.009 

199.946 

299.783 

399.459 

498.918 

598.106 

50 

.010 

199.939 

299.756 

399.389 

498.778 

597.862 

3 

100.011 

199.931 

299.726 

399.315 

498.630 

597.604 

10 

.013 

199.924 

299.695 

399.237 

498.474 

597.331 

20 

.014 

199.915 

299.662 

399.154 

498.309 

597.043 

30 

.015 

199.907 

299.627 

399.068 

498.136 

596.740 

40 

•017 

199.898 

299.591 

398.977 

497.955 

596.423 

50 

.019 

199.888 

299.553 

398.882  • 

497.765 

596.091 

4 

100.020 

199.878 

299.513 

398.782 

497.566 

595.744 

10 

.022 

199.868 

299.471 

398.679 

497.360 

595.383 

20 

.024 

199.857 

299.428 

398.571 

497.145 

595.007 

30 

.026 

199.846 

299.383 

398.459 

496.921 

594.617 

40 

.028 

199.834 

299.337 

398.343 

496.689 

594.212 

50 

.030 

199.822 

299.289 

398.223 

496.449 

593.792 

5 

100.032 

199.810 

299.239 

398.099 

496.201 

593.358 

10 

.034 

199.797 

299.187 

397.970 

495.944 

592.909 

20 

.036 

199.783 

299.134 

397.837 

495.678 

592.446 

30 

.038 

199.770 

299.079 

397.700 

495.405 

591.968 

40 

.041 

199.756 

299.023 

397.559 

495.123 

591.476 

50 

.043 

199.741 

298.964 

397.413 

494.832 

590.970 

6 

100.046 

199.726 

298.904 

397.264 

494.534 

590.449 

10 

.048 

199.710 

298.843 

397.110 

494.227 

589.913 

20 

.051 

199.695 

298.779 

396.952 

493.912 

589.364 

30 

.054 

199.678 

298.714 

396.790 

493.588 

588.800 

40 

.056 

199.662 

298.648 

396.623 

493.257 

588.221 

50 

.059 

199.644 

298.579 

396.453 

492.917 

587.628 

7 

100.062 

199.627 

298.509 

396.273 

492.568 

587.021 

10 

.065 

199.609 

298.438 

396.099 

492.212 

586.400 

20 

.068 

199.591 

298.364 

395.916 

491.847 

585.765 

30 

.071 

199.572 

298.289 

395.729 

491.474 

585.115 

40 

.075 

199.553 

298.212 

395.538 

491.093 

584.451 

50 

.078 

199.533 

298.134 

395.342 

490.704 

583.773 

8 

100.081 

199.513 

298.054 

395.142 

490.306 

583.081 

10 

.085 

199.492 

297.972 

394.938 

489.900 

582.375 

20 

.088 

199.471 

297.888 

394.731 

489.486 

581.654 

30 

.092 

199.450 

297.803 

394.518 

489.064 

580.920 

40 

.095 

199.428 

297.716 

394.302 

488.634 

580.172 

50 

.099 

199.406 

297.628 

394.082 

488.196 

579.409 

9 

100.103 

199.383 

297.538 

393.857 

487.749 

578.633 

10 

.107 

199.360 

297.446 

393.629 

487.294 

577.843 

20 

.111 

199.337 

297.352 

393.396 

486.832 

577.039 

30 

.115 

199.313 

297.257 

393.159 

486.361 

576.222 

40 

.119 

199.289 

297.160 

392.918 

485.882 

575.390 

50 

.123 

199.264 

297.062 

392.673 

485.395 

574.545 

10 

100.127 

199.239 

296.962 

392.424 

484.900    |    573.686 

173 


TABLE  V.— LONG  CHORDS. 


Degree 
of 
Curve. 

Actual 
Arc, 
One 
Station. 

LONG  CHORDS. 

2 

Stations. 

3 

Stations. 

4 
Stations. 

5 

Stations. 

6 

Stations. 

10°  10' 

100.131 

199.213 

296.860 

392.171 

484.397 

572.813 

20 

.136 

199.187 

396.756 

391.914 

483.886 

571.926 

30 

140 

199.161 

296.651 

391.652 

483.367 

571.027 

40 

.145 

199.134 

296.544 

391.387 

482.840 

570.113 

50 

.149 

199.107 

296.436 

391.117 

482.305 

569.186 

11 

100.154 

199.079 

296.325 

390.843 

481.762 

568.245 

10 

.158 

199.051 

296.214 

390.565 

481.211 

567.292 

20 

.163 

199.023 

296.100 

390.284 

480.653 

566.324 

30 

.168 

198.994 

295.985 

389.998 

480.086 

565.343 

40 

.173 

198.964 

295.868 

389.708 

479.511 

564.349 

50 

.178 

198.935 

295.750 

389.414 

478.929 

563.341 

12 

100.183 

198.904 

295.629 

389.116 

478.338 

562.321 

10 

.188 

198.874 

295.508 

388.814 

477.740 

561.287 

20 

.193 

198.843 

295.384 

388.508 

477.135 

560.240 

30 

.199 

198.811 

295.259 

388.197 

476.521 

559.180 

40 

.204 

198.779 

295.132 

387.883 

475.899 

558.107 

50 

.209 

198.747 

295.004 

387.565 

475.270 

557.020 

13 

100.215 

198.714 

294.874 

387.243 

474.633 

555.921 

10 

•  220 

198.681 

294.742 

386.916 

473.988 

554.809 

20 

•226 

198.648 

294.609 

386.586 

473.336 

553.684 

30 

.232 

198.614 

294.474 

386.252 

472.675 

552.546 

40 

.237 

198.579 

294.337 

385.914 

472.007 

551.395 

50 

.243 

198.544 

294.199 

385.572 

471.332 

550.232 

14 

100.249 

198.509 

294.059 

385.225 

470.649 

549.056 

10 

•255 

198.474 

293.918 

384.875 

469.958 

547.867 

20 

.261 

198.437 

293.774 

384.521 

469.260 

546.666 

30 

.267 

198.401 

293.629 

384.163 

468.554 

545.452 

40 

.274 

198.364 

293.483 

383.801 

467.840 

544.226 

50 

.280 

198.327 

293.335 

383.435 

467.119 

542.987 

15 

100.286 

198.289 

293.185 

383.065 

466.390 

541.736 

10 

•292 

198.251 

293.034 

382.691 

465.654 

540.472 

20 

•299 

198.212 

292.881 

382.313 

464.911 

539.196 

30 

306 

198.173 

292.726 

381.931 

464.160 

537.908 

40 

312 

198.134 

292.570 

381.546 

463.401 

536.608 

50 

.319 

198.094 

292.412 

381.156 

462.635 

535.296 

IG; 

100.326 

198.054 

292.252 

380.763 

461.862 

533.972 

10 

.333 

198.013 

292.091 

380.365 

461.081 

532.635 

20 

.339 

197.972 

291.928 

379.964 

460.293 

531.287 

30 

.346 

197.930 

291.764 

379.559 

459.498 

529.927 

40 

.353 

197.888 

291.598 

379.150 

458.695 

528.555 

50 

.361 

197.846 

291.430 

378.737 

457.886 

527.171 

17 

100.368 

197.803 

291.261 

378.320 

457.069 

525.778 

10 

.375 

197.760 

291.090 

377.900 

456.244 

524.369 

20 

.382 

197.716 

290.918 

377.475 

455.413 

522.950 

30 

.390 

197.672 

290.743 

377.047 

454.574 

521.519 

40 

.397 

197.628 

290.568 

376.615 

453.728 

520.073 

50 

.405 

197.583 

290.390 

376.179 

452.875 

518.625 

18 

100.412 

197.538 

290.211 

375.739 

452.015 

517.160 

10 

.420 

197.492 

290.031 

375.295 

451.147 

515.685 

20 

.428 

197.446 

289.849 

374.848 

450.373 

514.198 

30 

.436 

197.399 

£89.665 

374.397 

449.392 

512.699 

40 

.444 

197.352 

289.479 

373.942 

448.504 

511.190 

50 

A52 

197.305 

289.292 

373.483 

447.608 

509.670 

19 

100.4GO 

197.256 

289.104 

373.021 

446.706 

508.139 

10 

.468 

197.209 

288.913 

372.554 

445.797 

506.597 

20 

.476 

197.160 

288.722 

372.084 

444.881 

505.043 

30 

.484 

197.111 

288.528 

371.610 

443.957 

503.479 

40 

.493 

197.062 

288.333 

371.133 

443.028 

501.905 

50 

.501 

197.012 

288.137 

370.652 

442.091 

500.320 

20 

100.510 

196.963 

287.939 

370.167 

441.147 

498.724 

174 


TABLE  VI.-MID-ORDINATES  TO  LONG  CHORDS. 


Degree 
of 
Curve. 

1 

Station. 

a 

Stations. 

3 

Stations. 

4 

Stations. 

5 

Stations. 

6 

Stations. 

0°  l</ 

.036 

.145 

.327 

.582 

.909 

1.309 

20 

.073 

.291 

.654 

1.164 

1.818 

2.618 

80 

.109 

.436 

.982 

1.745 

2.727 

3.926 

40 

.145 

.582 

1.309 

2.327 

3.636 

5.235 

50 

183 

.727 

1.636 

2.909 

4.545 

6.544 

1 

218 

.873 

1.963 

3.490 

5.453 

7.852 

10 

255 

1.018 

2.291 

4.072 

6.362 

9.160 

20 

231 

1.164 

2.618 

4.654 

7.270 

10.468 

30 

.327 

1.309 

2.945 

5.235 

8.179 

11.775 

40 

364 

1.454 

3.272 

5.816 

9.087 

13.082 

50 

.400 

1.600 

3.599 

6.398 

9.994 

14.389 

2 

436 

1.745 

3.926 

6.979 

10.902 

15.694 

10 

473 

1.891 

4.253 

7.560 

11.809 

17.000 

20 

.509 

2.036 

4.580 

8.141 

12.716 

18.304 

30 

.545 

2.181 

4.907 

8.722 

13.623 

19.608 

40 

.582 

2.327 

5.234 

9.303 

14.529 

20.912 

50 

.618 

2.472 

5.561 

9.883 

15.435 

22.214 

3 

.654 

2.618 

5.888 

10.464 

16.341 

23.516 

10 

.691 

2.763 

6.215 

11.044 

17.246 

24.817 

20 

.727 

2.908 

6.542 

11.624 

18.151 

26.117 

30 

.763 

3.054 

6.868 

12.204 

19.055 

27.416 

40 

.800 

3.199 

7.195 

12.784 

19.959 

28.714 

50 

.836 

3.345 

7.522 

13.363 

20.863 

30.C12 

4 

.872 

3.430 

7.848 

13.943 

21.766 

31.308 

10 

.009 

3.635 

8.175 

14.522 

22.668 

32.6(3 

20 

.945 

3.781 

8.501 

15.101 

23.570 

83.893 

30 

.983 

3.  920 

8.828 

15.680 

24.471 

35.189 

40 

.018 

4.071 

9.154 

16.258 

25.372 

36.480 

50 

.054 

4.217 

9.480 

16.837 

26.272 

37.770 

5 

.091 

4.362 

9.807 

17.415 

27.171 

39.053 

10 

.127 

4.507 

10.133 

17.992 

28.070 

40.346 

20 

.164 

4.653 

10.459 

18.570 

28.968 

41.631 

30 

.200 

4.798 

10.785 

19.147 

29.866 

42.916 

40 

.237 

4.943 

11.111 

19.724 

30.762 

44.198 

50 

.273 

5.088 

11.436 

20.301 

31.658 

45.479 

6 

.309 

5.234 

11.762 

20.877 

32.553 

46.759 

10 

.346 

5.379 

12.088 

21.453 

33.448 

48.037 

20 

.382 

5.524 

12.413 

22.029 

34.341 

"49.313 

30 

1.418 

5.663 

12.739 

22.604 

35.234 

50.587 

40 

1.453 

5.814 

13.064 

23.179 

36.126 

51.860 

50 

1.491 

5.960 

13.389 

23.754 

37.017 

53.130 

7 

1.523 

6.105 

13.715 

24.328 

37.907 

54.399 

10 

1.564 

6.250 

14.040 

24.902 

38.796 

55.660 

20 

1.600 

6.395 

14.365 

25.476 

39.684 

56.931 

30 

1.637 

6.540 

14.689 

26.049 

40.571 

58.193 

40 

1.673 

6.683 

15.014 

26.622 

41.458 

59.451 

50 

1.710 

6.831 

15.339 

27.195 

42.343 

60.712 

8 

1.746 

6.976 

15.663 

27.767 

43.227 

61.969 

10 

1.782 

7.121 

15.988 

28.338 

44.110 

63.223 

20 

1.819 

7.266 

16.312 

28.910 

44.992 

64.475 

30 

1.855 

7.411 

16.636 

29.481 

45.873 

65.724 

40 

1.892 

7.556 

16.960 

30.051 

46.753 

66.972 

50 

1.928 

7.701 

17.284 

30.621 

47.632 

68.216 

9 

1.965 

7.846 

17.608 

31.190 

48.510 

69.459 

10 

2.001 

7.991 

17.932 

31.759 

49.386 

70.699 

20 

2.037 

8.136 

18.255 

32.328 

50.261 

71.936 

30 

2.074 

8.281 

18.578 

32.896 

51.135 

73.171 

40 

2.110 

8.426 

18.902 

33.464 

52.008 

74.403 

50 

2.147 

8.571 

19.225 

34.031 

52.880 

75.632 

10 

2.183 

8.716 

19.548 

34.597 

53.750 

76.859 

TABLE  VI.— MID-ORDINATES  TO  LONG  CHORDS. 


Degree 
of 
Curve. 

1 

Station. 

» 

Stations. 

3 

Stations. 

4 

Stations. 

5 

Stations. 

6 

Stations. 

10°  10' 

2.219 

8.860 

19.870 

35.164 

54.619 

78.083 

20 

2.256 

9.005 

20.193 

35.729 

55.486 

79.305 

30 

2.293 

9.150 

20.516 

36.294 

56.353 

80.523 

40 

2.329 

9.295 

20.838 

36.859 

57.218 

81.739 

50 

2.365 

9.440 

21.160 

37.423 

58.081 

82.951 

11 

2.402 

9.585 

21.483 

37.986 

58.943 

84.161 

10 

2.438 

9.729 

21.804 

38.549 

59.804 

85.368 

20 

2.475 

9.874 

22.126 

39.111 

60.663 

86.571 

80 

2.511 

10.019 

22.448 

39.673 

61.521 

87.772 

40 

2.547 

10.164 

22.7'69 

40.234 

62.377 

88.969 

50 

2.584 

10.308 

23.090 

40.795 

63.232 

90.164 

12 

2.620 

10.453 

23.412 

41.355 

64.085 

91.355 

10 

'2.657 

10.597 

23.732 

41.914 

64.937 

92.542 

20 

2.693 

10.742 

24.053 

42.473 

65.787 

93.727 

30 

2.730 

10.887 

24.374 

43.031 

66.636 

94.908 

40 

2.766 

11.031 

24.694 

43.588 

67.482 

96.086 

50 

2.803 

11.176 

25.014 

44.145 

68.328 

97.260 

13 

2.839 

11.320 

25.334 

44.701 

69.171 

98.431 

10 

2.876 

11.465 

25.654 

45.256 

70.013 

99.598 

20 

2.912 

11.609 

25.974 

45.811 

70.854 

100.762 

30 

2.949 

11.754 

26.293 

46.365 

71.692 

101  .  922 

40 

2.985 

11.898 

26.612 

46.919 

72.529 

103.  Or,  9 

50 

3.022 

12.043 

26.931 

47.472 

73.364 

104.232 

14 

3.058 

12.187 

27.250 

48.024 

74.197 

105.  S81 

10 

3.095 

12.331 

27.569 

48.575 

75.029 

106.527 

20 

3.131 

12.476 

27.887 

49.126 

75.859 

107.  6C9 

30 

3.168 

12.620 

28.206 

49.676 

76.687 

108.807 

40 

3.204 

12.764 

28.524 

50.225 

77.513 

109.941 

50 

3.241 

12.908 

28.841 

50.773 

78.337 

111.071 

15 

3.277 

13.053 

29.159 

51.321 

79.159 

112.197 

10 

3.314 

13.197 

29.476 

51.868 

79.979 

113.319 

20 

3.350 

13.341 

29.794 

52.414 

80.798 

114.4^8 

30 

3.387 

13.485 

30.111 

52.959 

81.614 

115.552 

40 

3.423 

13.629 

30.427 

53.504 

82.429 

116.662 

50 

3.460 

13.773 

30.744 

54.048 

83.241 

117.768 

16 

3.496 

13.917 

31.060 

54.591 

84.052 

118.870 

10 

3.533 

14.061 

31.376 

55.133 

84.861 

119.967 

20 

3.569 

14.205 

31.692 

55.675 

85.667 

121.061 

30 

3.606 

14.349 

32.008 

56.215 

86.471 

122.150 

40 

3.643 

14.493 

32.323 

56.755 

87.274 

123.  ££5 

50 

3.679 

14.637 

32.638 

57.294 

88.074 

124.315 

17 

3.716 

14.781 

32.953 

57.832 

88.872 

125.  S91 

10 

3.752 

14.925 

33.267 

58.369 

89.C68 

126.463 

20 

8.789 

15.069 

33.582 

58.906 

90.462 

127.  E30 

30 

3.825 

15.212 

33.896 

59.441 

91.254 

128.593 

40 

3.862 

15.356 

34.210 

59.976 

92.043 

129.  GC1 

50 

£.899 

15.500 

34.523 

60.510 

92.830 

130.704 

18 

3.935 

15.643 

34.837 

61.042 

93.616 

131.753 

10 

3.972 

15.787 

35.150 

61.574 

94.398 

132.797 

20 

4.008 

15.931 

35.463 

62.106 

95.179 

133.837 

30 

4.045 

16.074 

35.775 

62.636 

95.957 

134.872 

40 

4.081 

16.218 

36.088 

63.165 

96.733 

135.902 

50 

4.118 

16.361 

36.400 

63.693 

97.506 

136.928 

19 

4.155 

16.505 

36.712 

64.221 

98.278 

137.948 

10 

4.191 

16.648 

37.023 

64.747 

99.047 

138.964 

20 

4.228 

16.792 

37.  £M 

65.273 

99.813 

139.975 

30 

4.265 

16  935 

37.645 

65.797 

100.577 

140.981 

40 

4.301 

17.078 

37.956 

66.321 

101.339 

141.982 

50 

4.338 

17.222 

38.266 

66.843 

102.098 

142.978 

20 

4.374 

17.365          38.576 

67.365 

102.855 

143.969 

TABLE  VII. -MINUTES  IN  DECIMALS  OF  A  DEGREE. 


t 

0" 

10" 

15' 

20" 

30" 

40" 

45' 

50' 

/ 

0 

.00000 

00278 

.00417 

.00556 

.00833 

.01111 

.01250 

.01389 

0 

1 

.01667 

.01944 

.02083 

.02222 

.02500 

.02778 

.02917 

.03055 

1 

2 

.03333 

.03611 

.03750 

.03889 

.04167 

.04444 

.04583 

.04722 

2 

3 

.05000 

.05278 

.05417 

.05556 

.05833 

.06111 

.06250 

.06389 

3 

4 

.06667 

.06944 

.07083 

.07222 

.07500 

.07778 

.07917 

.08056 

4 

5 

.08333 

.08611 

.08750 

.08889 

.09167 

.09444 

.09583 

.09722 

5 

6 

.10000 

.10278 

.10417 

.10556 

.10833 

.11111 

.11250 

.11389 

6 

7 

.11667 

.11944 

.12083 

.12222 

.12500 

.12778 

.12917 

.13056 

7 

8 

.13333 

.13611 

.13750 

.13889 

.14167 

.14444 

.14583 

.14722 

8 

9 

.15000 

.15278 

.15417 

.15556 

.15833 

.16111 

.16250 

.16389 

9 

10 

.16667 

.16944 

.17083 

.17222 

.17500 

.17778 

.17917 

.18056 

10 

11 

.18333 

..18611 

.18750 

.18889 

.19167 

.19444 

.19583 

.19722 

11 

12 

.20000 

.20278 

.20417 

.20556 

.20833 

.21111 

.21250 

.21389 

12 

13 

.21667 

.21944 

.22083 

.22222 

.22500 

.22778 

.22917 

.23056 

13 

14 

.23333 

.23611 

.23750 

.23889 

.24167 

.24444 

.24583 

.24722 

14 

15 

.25000 

.25278 

.25417 

.25556 

.25833 

.26111 

.26250 

.26389 

15 

16 

.26067 

.26944 

.27083 

.27222 

.27500 

.27778 

.27917 

.28056 

16 

17 

.28333 

.28611 

.28750 

.28889 

.29167 

.29444 

.29583 

.29722 

17 

18 

.30000 

.30278 

.30-117 

.30556 

.30833 

.31111 

.31250 

.31389 

18 

19 

.31667 

.31944 

.32083 

.32222 

.32500 

.32778 

.32917 

.33056 

19 

20 

.33333 

.33611 

.33750 

.33889 

.34167 

.34444 

.34583 

.34722 

20 

21 

.35000 

.35278 

.35417 

.35556 

.35833 

.36111 

.36250 

.36389 

21 

22 

.36667 

.36944 

.37033 

.37222 

.37500 

.37778 

.37917 

.38056 

22 

23 

.38333 

.38611 

.38750 

.38389 

.39167 

.39444 

.39583 

39722 

23 

24 

.40000 

.40278 

.40417 

.40556 

.40833 

.41111 

.41250 

.41389 

24 

25 

.41667 

.41944 

.42083 

.42222 

.42500 

.42778 

.42917 

.43056 

25 

26 

.43333 

.43611 

.43150 

.43389 

.44167 

4441'! 

.44583 

.44722 

26 

27 

.45000 

.45278 

.45417 

.45556 

.45833 

146111 

.46250 

.46389 

27 

28 

.46667 

.46944 

.47083 

.47222 

.47500 

.47778 

.47917 

.48056 

28 

29 

.483.33 

.48611 

.48750 

.48889 

.49167 

.49444 

.49583 

.49722 

29 

30 

.50000 

.50278 

.50417 

.50556 

.50833 

.51111 

.51250 

.51389 

30 

31 

.51667 

.51944 

.52083 

.52222 

.52500 

.52778 

.E2917 

.53056 

31 

32 

.53333 

.53611 

.53750 

.53839 

.54167 

.54444 

.54583 

.54722 

32 

33 

.55000 

.55278 

.55417 

.55556 

.55a33 

.56111 

.56250 

.56389 

33 

34 

.56667 

.56944 

.57083 

.57222 

.57500 

.57778 

.57917 

.58056 

34 

35 

.58333 

.58611 

.58750 

.53889 

.59167 

.59444 

.59583 

.59722 

35 

36 

.60000 

.60278 

.60417 

.60556 

.60833 

.61111 

.61250 

.61389 

36 

37 

.61667 

.61944 

.62083 

.62222 

.62500 

.62778 

.62917 

.63056 

37 

38 

.63333 

.63611 

.637'50 

.63889 

.64167 

.64444 

.64583 

.64722 

38 

39 

.65000 

.65278 

.65417 

.65556 

.65833 

.66111 

.66250 

.66389 

39 

40 

.66667 

.66944 

.67083 

.67222 

.67500 

.67778 

.67917 

.68056 

40 

41 

.68333 

.68611 

.68750 

.68889 

.69167 

.69444 

.69583 

.69722 

41 

42 

.70000 

.70278 

.70417 

.70556 

.70833 

.71111 

.71250 

.71389 

42 

43 

.71667 

.71944 

.72083 

72222 

.72500 

.72778 

.72917 

.73056 

43 

44 

.73333 

.73611 

.73750 

!73889 

.74167 

.74444 

.74583 

.74722 

44 

45 

.75000 

.75278 

.75417 

.75556 

.75833 

.76111 

.76250 

.76389 

45 

46 

.76667 

.76944 

.77083 

.77222 

.77500 

.77778 

.77917 

.78056 

46 

47 

.78333 

.78011 

.78750 

.78889 

.79167 

.79444 

.79583 

.79722 

47 

48 

.8COOO 

.80278 

.80417 

.80556 

.80833 

.81111 

.81250 

.81389 

48 

49 

.81667 

.81944 

.82083 

.82222 

.82500 

.82778 

.82917 

.83056 

49 

50 

.83333 

.83611 

.83750 

.83889 

.84167 

.84444 

.84583 

.84722 

50 

51 

.85000 

.85278 

.85417 

.85556 

.85833 

86111 

.86250 

.86389 

51 

52 

.86667 

.86944 

.87083 

.87222 

.87500 

.87778 

.87917 

.88056 

52 

53 

.88333 

.88611 

.88750 

.88889 

.89167 

.89444 

.89583 

.89722 

53 

54 

.90000 

.90278 

.90417 

.90556 

.90833 

.91111 

.91250 

.91389 

54 

55 

.91667 

.91944 

.92083 

.92222 

.92300 

.92778 

.92917 

.93056 

55 

56 

.93333 

.93611 

.93750 

.93889 

.94167 

.94444 

.94583 

.94722 

56 

57 

.95000 

.95278 

.95417 

.95556 

.95833 

.96111 

.96250 

.96389 

57 

58 

.90667 

.96944 

.97083 

.97222 

.97500 

.97778 

.97917 

.98056 

58 

59 

.98333 

.98611 

.98750 

.98889 

.99167 

.99444 

.99583 

.99722 

59 

/ 

0" 

10" 

15' 

20" 

30" 

40" 

45" 

50" 

/ 

TABLE  VIII.— SQUARES,  CUBES,  SQUARE  ROOTS,  AND  CUBE  ROOT 


No. 

Squares. 

Cubes. 

Xa£   |  CubeB«*. 

Reciprocals. 

1 

1 

1 

1.0000000 

1.0000000 

1.000000000 

2 

4 

8 

1.4142136 

1.2599210 

.500000000 

3 

9 

27 

1.7320508 

1.4422496 

.333333333 

4 

16 

64 

2.0000000 

1.5874011 

.250000000 

5 

25 

125 

2.2360680 

1.7099759 

.200000000 

6 

36 

216 

2  4494897 

1.8171206 

.166666667 

7 

49 

343 

2.6457513 

1.9129312 

.142857143 

8 

64 

512 

2.8284271 

2.0000000 

.125000000 

9 

81 

729 

3.0000000 

2.0800837 

.111111111 

10 

100 

1000 

3.1622777 

2.1544347 

.100000000 

11 

121 

1331 

3.3166248 

2.2239801 

.090909091 

12 

144 

1728 

3  4641016 

2.2894286 

.083333333 

13 

169 

2197 

3.6055513 

2.3513347 

.076923077 

14 

196 

2744 

3.7416574 

2.4101422 

.071428571 

15 

225 

3375 

3.8729833 

2.4662121 

.066666667 

16 

256 

4096 

4.0000000 

2.5198421 

.062500000 

17 

289 

4913 

4.1231056 

2.5712816 

.058823529 

18 

324 

5832 

4.2426407 

2.6207414 

.055555556 

19 

361 

6859 

4.3588989 

2.6684016 

.052631579 

20 

400 

8000 

4.4721360 

2.7144177 

.050000000 

21 

441 

9261 

4.5825757 

2.7589243 

.047619048 

22 

484 

10648 

4.6904158 

2.8020393 

.045454545 

23 

529 

12167 

4.7958315 

2.8438670 

.043478261 

24 

576 

13824 

4.8989795 

2.8844991 

.041666667 

25 

625 

15625 

5.0000000 

2.9240177 

.040000000 

26 

676 

17576 

5.0990195 

2.9624960 

.038461538 

27 

729 

19683 

5.1961524 

3.0000000 

.037037037 

28 

784 

21952 

5.2915026 

3.0365889 

.035714286 

29 

841 

24389 

5.3851648 

3.0723168 

.034482759 

30 

900 

27000 

5.4772256 

3.1072325 

.033333333 

31 

961 

29791 

5.5677644 

3.1413806 

.032258065 

32 

1024 

32768 

5.6568542 

5.1748021 

.031250000 

33 

1089 

35937 

5.7445626 

8.2075343 

.030303030 

34 

1156 

39304 

5.8309519 

3.2396118 

.029411765 

35 

1225 

42875 

5.9160798 

3.2710663 

.028571429 

36 

1296 

46656 

6.0000000 

3.3019272 

.027777778 

37 

1369 

50653 

6.0827625 

3.3322218 

.027027027 

38 

1444 

54872 

6.1644140 

3.3619754 

.026315789 

39 

1521 

59319 

6.2449980 

3.3912114 

.025641026 

40 

1600 

64000 

6.3245553 

3.4199519 

.025000000 

41 

1681 

68921 

6.4031242 

3.4482172 

.024390244 

42 

1764 

74088 

6.4807407 

3.4760266 

.023809524 

43 

1849 

79507 

6.5574385 

3.5033981 

.023255814 

44 

1936 

85184 

6.6332496 

3.5303483 

.022727273 

45 

2025 

91125 

6.7082039 

3.5568933 

.022222222 

46 

2116 

97336 

6.7823300 

3.5830479 

.021739130 

47 

2209 

103823 

6.8556546 

3.G088261 

.021276600 

48 

2304 

110592 

6.9282032 

3.6342411 

.020833333 

49 

2401 

117649 

7.0000000 

3.6593057 

.020408163 

50 

2500 

125000 

7.0710678 

3.6840314 

.020000000 

51 

2601 

132651 

7.1414284 

3.7084298 

.019607843 

52 

2704 

140608 

7.2111026 

3.7325111 

.019230769 

53 

2809 

148877 

7.2801099 

3.7562858 

.018867925 

54 

2916 

157464 

7.3484692 

3.7797631 

,0185ia519 

55 

3025 

166375 

7.4161985 

3.8029525 

.018181818 

56 

3136 

175616 

7.4833148 

3.8258624 

.017857143 

57 

3249 

185193 

7.5498344 

3.8485011 

.017543860 

58 

3364 

195112 

7.6157731 

3.8708766 

.017241379 

59 

3481 

205379 

7.6811457 

3.8929965 

.016949153 

60 

3600 

216000 

7.7459667 

3.9148676 

.016666667 

61 

3721 

'  226981 

7.8102497 

3.9364972 

.016393443 

62 

3844 

238328 

7.8740079 

3.9578915 

.016129032 

TABLE  VIII.- Continued. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

63 

3969 

250047 

7.9372539 

3.9790571 

.015873016 

64 

4096 

262144 

8.0000000 

4.0000000 

.015625000 

65 

4225 

274025 

8.0622577 

4.0207256 

.015384615 

68 

4356 

287496 

8.1240384 

4.0412401 

.  .015151515 

67  1 

4489 

300763 

8.1853528 

4.0615480 

.014925373 

68 

46.24 

314432 

8.2462113 

4.0816551 

.014705882 

69 

4761 

328509 

8.3066239 

4.1015661 

.014492754 

70 

4900 

343000 

8.3666003 

4.1212853 

.014285714 

71 

5041 

357911 

8.4261498 

4.1408178 

.014084507 

72 

5184 

373248 

8.4852814 

4.1601676 

.013888889 

73 

5329 

389017 

8.5440037 

4.1793390 

.013698630 

74 

5476 

405224 

8.6023253 

4.1983364 

.013513514 

75 

5625 

421875 

8.6602540 

4.2171633 

.013333333 

76 

5776 

438976 

8,7177979 

4.2358236 

.013157895 

77 

5929 

456533 

8*  7749644 

4.2543210 

.012987013 

78 

6084 

474552 

8.8317609 

4.2726586 

.012820513 

79 

6241 

493039 

8.8881944 

4.2908404 

.012658228 

80 

6400 

512000 

8.9442719 

4.3088695 

.012500000 

81 

G5G1 

531441 

9.0000000 

4.3267487 

.012345679 

82 

6724 

551368 

9.0553851 

4.3444815 

.012195122 

83 

6889 

571787 

9.1104336 

4.3620707 

.012048193 

84 

7056 

592704 

9.1651514 

4.3795191 

.011904762 

85 

7225 

614125 

9.2195445 

4.3968296 

.011764706 

86 

7396 

636056 

9.2736185 

4.4140049 

.011627907 

87 

7569 

658503 

9.327'3791 

4.4310476 

.011494253 

88 

7744 

681472 

9.3808315 

4.4479602 

.011363636 

89 

7921 

704969 

9.4339811 

'4.4&47451 

.011235955 

90 

8100 

729000 

9.4868330 

4.4814047 

.011111111 

91 

8281 

753571 

9.5393920 

4.4979414 

.010989011 

92 

8464 

778683 

9.5916630 

4.5143574 

.010869565 

93 

8649 

804357 

9.6436508 

4.5306549 

.010752688 

94 

8836 

830584  • 

9.6953597 

4.5468359 

.010638298 

95 

9025 

857375 

9.74679-43 

4.5629026 

.010526316 

96 

9216 

884736 

9.7979590 

4.5788570 

.010416667 

97 

9409 

912673 

9.8488578 

4.5947009 

.010309278 

J98 

9604 

941192 

9.8904949 

4.6104363 

.010204082 

99 

9801 

970299 

9.9498744 

4.0260G50 

.010101010 

100 

10000 

1000000 

10.0000000 

4.6415888 

.010000000 

101 

10201 

1030301 

10.0498756 

4.6570095 

.009900990 

102 

10404 

1061208 

10.0995049 

4.6723287 

.009803922 

103 

10609 

1092727 

10.1488916 

4.6875482 

.009708738 

104 

10816 

1124864 

10.1980390 

4.7026694 

.009615385 

105 

11025 

1157625 

10.2469508 

4.7176940 

.009523810 

106 

11236 

1191016 

10.2956301 

4.7326235 

.009433962 

107 

11449 

1225043 

10.3440804 

4.7474594 

.009345794 

108 

11G64 

1259712 

10.3923048 

4.7622032 

.003259259 

109 

11881 

1295029 

10.4403065 

4.7768562 

.009174312 

110 

12100 

1331000 

10.4880885 

4.7914199 

.009090909 

111 

12321 

1367631 

10.5356538 

4.8058955 

.009009009 

112 

12544 

1404928 

10.5830052 

4.8202845 

.008928571 

113 

12769 

1442897 

10.6301458 

4.8345881 

.008849558 

114 

12996 

1481544 

10.  €770783 

4.848S076 

.008771930 

115 

13225 

1520875 

10.7238053 

4.8629442 

.008695652 

116 

13456 

1560896 

10.7703296 

4.8769990 

.008620690 

117 

13689 

1601613 

10.8166538 

4.8909732 

.008547009 

118 

13924 

1643032 

10.8627805 

4.9048681 

.008474576 

119 

14161 

1685159 

10.9087131 

4.9186847 

.008403361 

120 

14400 

1728000 

10.9544513 

4.9324242 

.008333333 

121 

14641 

1771561 

1  1.00.  -0000 

4.9460874 

.008264463 

132 

14884 

1815848 

11.0453610 

4.9596757 

.008196721 

123 

15129 

1860867 

11.0905365 

4.9731898 

.008130081 

124 

15376 

1906624 

11.1355287 

4.9866310 

.008064516 

179 


TABLE  VIII.— Continued. 


~  NO. 

Squares. 

Cubes. 

Square 
Boots. 

Cube  Roots. 

. 
Reciprocals^ 

125 

15625 

1953125 

11.1803399 

5.0000000 

.008000000 

126 

15876 

2000376 

11.2249722 

5.0132979 

.007936508 

127 

16129 

2048383 

11.2694277 

5.0265257 

.007874016 

128 

16384 

2097152 

11.3137085 

5.0396842 

.007812500 

129 

16641 

2146689 

11.3578167 

5,0527743 

.007751938 

130 

16900 

2197000 

11.4017543 

5.0657970 

.007692308 

131 

17161 

2248091 

11.4455231 

5.0787531 

.007633588 

132 

17424 

2299968 

11.4891253 

5.0916434 

.007575758 

133 

17689 

2352637 

11.5325626 

5.1044687 

.007518797 

134 

17956 

2406104 

11.5758369 

5.1172299 

.007462687 

135 

18225 

2460375 

11.6189500 

5.1299278 

.007407407 

136 

18496 

2515456 

11.6619038 

5.1425632 

.007352941 

137 

18769 

2571353 

11.7046999 

5.1551367 

.007299270 

138 

19044 

2628072 

11.7473401 

5.1676493 

.007246377 

139 

19321 

2685619 

11.7898261 

5.1801015 

.007194245 

140 

19600 

2744000 

11.8321596 

5.1924941 

.007142857 

141 

19881 

2803221 

11.8743421 

5.2048279 

.007092199 

142 

20164 

2863288 

11.  91637'53 

5.2171034 

.007042254 

143 

20449 

2924207 

11.9582607 

5.2293215 

.006993007 

144 

20736 

2985984 

12.0000000 

5.2414828 

.006944444 

145 

21025 

3018625 

12.0415946 

5.2535879 

.006896552 

146 

21316 

3112136 

12.0830460 

6.2656374 

.006849315 

147 

21609 

3176523 

12.1243557 

5.2776321 

.006802721 

148 

21904 

3241792 

12.1655251 

5.2895725 

.006756757 

149 

22201 

3307949 

12.2065556 

5.3014593 

.006711409 

150 

22500 

3375000 

12.2474487 

5.3132928 

.006666667 

151 

22801 

3442951 

12.2882057 

5.3250740 

.006622517 

152 

23104 

3511808 

12.3288280 

5.3368033 

.006578947 

153 

23409 

3581577 

12.3693169 

5.3484812 

.006535948 

,  154 

23716 

3652264 

12.4096736 

5.3601084 

.006493506 

155 

24025 

3723875 

12.4498996 

5.3716854 

.006451613 

156 

24336 

3796416 

12.48999CO 

5.3832126 

.006410256 

157 

24649 

3869893 

12.5299641 

5  3946907 

.006369427 

158 

24964 

3944312 

12.5698051 

6.4061202 

.006329114 

159 

25281 

4019679 

12.6095203 

5.4175015 

.006289308 

160 

25600 

4096000 

12.6491106 

5.4288353 

.006250000 

161 

25921 

4173281 

12.G885775 

5.4401218 

.COG211180 

162 

26244 

4251528 

12.7279221 

5.4513618 

.006172840 

1G3 

26569 

4330747 

12.7671453 

5.4625556 

.006134969 

164 

26896 

4410944 

12.80G2485 

5.4737037 

.006097561 

165 

27225 

4492125 

12.8452326 

5.4848066 

.006060606 

166 

27556 

4574296 

12.8840987 

5.4958647 

.006024096 

167 

27889 

4657463 

12.9228480 

6.50G8784 

.005988024 

168 

28224 

4741632 

12.9G14814 

5.5178484 

.005952381 

169 

28561 

4826809 

13.0000000 

5.5287748 

.005917160 

170 

28900 

4913000 

13.0384048 

5.5396583 

.005882353 

171 

29241 

5000211 

13.07G6968 

5.5504991 

.005847953 

172 

29584 

5088448 

13.1148770 

5.5612978 

.005813953 

173 

29929 

5177717 

13.1529464 

5.5720546 

.005780347 

174 

30276 

5268024 

13.1909060 

5.5827702 

.005747126 

17'5 

30625 

5359375 

13.2287566 

5.5934447 

.005714286 

176 

30976 

5451776 

13.2664992 

5.G040787 

.005681818 

177 

31329 

5545233 

13.3041347 

5.6146724 

.005649718 

178 

31684 

5639752 

13.3416641 

6.6252263 

.005617978 

179 

32041 

5735339  . 

13.3790882 

5.6357408 

.005586592 

180 

32400 

5832000 

13.4164079 

5.6462162 

.005555556 

181 

32761 

5929741 

13.4536240 

5.6566528 

.005524862 

182 

33124 

6028568 

13.4907376 

5.G670511 

.005494505 

183 

33489 

6128487 

13.5277493 

5.6774114 

.005464481 

184 

33856 

6229504 

13.5646600 

5.6877340 

.005434783 

"yJ85 

34225 

6331625 

13.6014705 

5.6980192 

.005405405 

186 

34596 

6434856 

13.6381817 

6.7082675 

.005376344 

180 


TABLE  VlU.-Continued. 


No.  1  Squares. 

Cubes. 

Square 
Roots. 

Cube  Boots. 

Reciprocals. 

187 

34969 

6539203 

13.6747943 

5.7184791 

.005347594 

188 

35344 

6644672 

13.7113092 

5.7286543 

.005319149 

189 

35721 

6751269 

13.7477271 

5.7387936 

.005291005 

190 

36100 

6859000 

13.7840488 

5.7488971 

.005263158 

191 

36481 

6967871 

13.8202750 

5.7589652 

.005235602 

192 

36864 

7077888 

13.8564065 

5.7689982 

.005208333 

193 

37249 

7189057 

13.8924440 

5.7789966 

.005181347 

194 

37636 

7301384 

13.9283883 

5.7889604 

.005154639 

195 

38025 

7414875 

13.9642400 

5.7988900 

.005128205 

196 

38416 

7529536 

14.0000000 

5.8087857 

.005102041 

197 

38809 

7645373 

14.0356688 

5.8186479 

.005076142 

198 

39204 

7762392 

14.0712473 

5.8284767 

.005050505 

199 

39601 

7880599 

14.1067360 

5.8382725 

.005025126 

200 

40000 

8000000 

14.1421356 

5.8480355 

.005000000 

201 

40401 

8120601 

14.1774469 

5.8577660 

.004975124 

202 

40804 

8242408 

14.2126704 

5.8674643 

.004950495 

203 

41209 

8365427 

14.2478068 

5.8771307 

.004926108 

204 

41616 

8489664 

14.2828569 

5.8867653 

.004901961 

205 

42025 

8615125 

14.3178211 

5.8963685 

.004878049 

206 

42436 

8741816 

14.3527001 

5.9059406 

.004854369 

207 

42849 

8869743 

14.3874946 

5.9154817 

.004880918 

203 

43264 

8998912 

14.4222051 

5.9249921 

.004807692 

209 

43681 

9129329 

14.4568323 

6.9344721 

.004784689 

210 

44100 

9261000 

14.4913767 

5.9439220 

.004761905 

211 

44521 

S393931 

14.5258390 

5.95S3418 

.004739336 

212 

44944 

9528128 

14.5602198 

5.9627320 

.004716981 

213 

45369 

9663597 

14.5945195 

5.9720926 

.004694836 

214 

45796 

9800344 

14.6287388 

5.9814240 

.004672897 

215 

46225 

9938875 

14.6628783 

5.9907264 

.004651163 

216 

46656 

10077696 

14.6969385 

6.0000000 

.004629630 

217 

47089 

10218313 

14.7309199 

6.0092450 

.004608295 

218 

47524 

10360232 

14.7648231 

6.0184617 

.004587156 

219 

47961 

10503459 

14.7986486 

6.0276502 

.004566210 

220 

48400 

10648000 

14.8323970 

6.0368107 

.004545455 

221 

48841 

10793861 

14.8660687 

6.0459435 

.004524887 

222 

49284 

10941048 

14.8996644 

6.0550489 

.004504505 

223 

49729 

11089567 

14.9331845 

6.0641270 

.004484305 

224 

50176 

11239424 

14.9666295 

6.0731779 

.004464286 

225 

50625 

11390625 

15.0000000 

6.C822020 

.004444444 

226 

51076 

11543176 

15.0332964 

6.0911994 

:  004424779 

227 

51529 

11697083 

15.0665192 

6.1001702 

.004405S86 

228 

51984 

11852352 

15.0996689 

6.1091147 

.004385965 

229 

52441 

12008989 

15.1327460 

6.1180332 

.0043G6812 

230 

52900 

12167000 

15.1657509 

6.1269257 

.004347826 

231 

53361 

12326391 

15.1986842 

6.1  357  924 

.004329004 

232 

53824 

12487168 

15.2315462 

6.1446337 

.004310345 

233 

54289 

12649337 

15.2643375 

6.1534495 

.004291845 

234 

54756 

12812904 

15.2970585 

6.1622401 

.  00427  a504 

235 

55225 

12977875 

15.3297097 

6.1710058 

.004255319 

236 

55G96 

13144256 

15.3622915 

6.1797466 

.004237288 

237 

561G9 

13312053 

15.3948043 

6.1884628 

.004219409 

238 

56644 

13481272 

15.4272486 

6.1971544 

.004201681 

239 

57121 

13651919 

15.4596248 

6.2058218 

.004184100 

240 

57600 

13824000 

35.4919334 

6.2144C50 

.004166667 

241 

58081 

13997521 

15.5841747 

6.2230843 

.004149378 

242 

58564 

14172488 

15.5563492 

6.2316797 

.004132231 

243 

£9049 

14348907 

15.5884573 

6.2402515 

.004115226 

244 

595:36 

14526784 

15.6204994 

6.2487998 

.004098361 

245 

60025 

14706125 

15.6524758 

6.2573248 

.004081633 

246 

60516 

14886936 

15.6843871 

6.2658266 

.004065041 

247 

61009 

15069223 

15.7162336 

6.2743054 

.004048583 

248 

61504 

15252992 

15.7480157 

6.2827613 

.004032258 

181 


TABLE  VIII.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Boots. 

Cube  Boots. 

Reciprocals. 

249 

62001 

15438249 

15.7797338 

6.2911946 

.004016064 

250 

62500 

15625000 

15.8113883 

6.2996053 

.004000000 

251 

63001 

15813251 

15.8429795 

6.3079935 

.003984064 

252 

63504 

16003008 

15.8745079 

6.3163596 

.003968254 

253 

64009 

16194277 

15.9059737 

6.3247035 

.003952509 

254 

64516 

16387064 

15.9373775 

6.3330256 

.003937008 

255 

65025 

16581375 

15.9687194 

6.3413257 

.003921509 

256 

65536 

16777216 

16.0000000 

6.3496042 

.003906250 

257 

615049 

16974593 

16.0312195 

6.3578611 

.003891051 

258 

66564 

17173512 

16.0623784 

6.3600908 

.  00387590  J 

259 

67081 

17373979 

16.09347'G9 

6.3743111 

.00:3861004 

260 

67600 

17576000 

16.1245155 

6.3825043 

.003846154 

261 

68121 

17779581 

16.1554944 

6.3906765 

.003831418 

262 

68644 

17984728 

16.1864141 

6.3988279 

.003816794 

263 

69169 

18191447 

16.2172747 

6.4069585 

.003802281 

264 

69696 

18399744 

16.2480768 

6.4150G87 

.003787879 

265 

70225 

18609625 

16.2788206 

6.4231583 

.003773585 

266 

70756 

18821096 

16.3095064 

6.4312276 

.003759398 

267 

71289 

190:34163 

16.3401346 

6.4392767 

.003745318 

268 

718.24 

19248832 

16.3707055 

6.4473057 

.003731343 

269 

72361 

19465109 

16.4012195 

C.  4553148 

.003717472 

270 

72900 

19683000 

16.4316767 

6.4633041 

.003703704 

271 

73441 

19902511 

16.4620776 

6.4712736 

.003090037 

272 

73984 

20123648 

16.4924225 

6.4792236 

.003676471 

273 

74523 

20346417 

16.5227116 

6.4871541 

.003663004 

274 

75076 

20570824 

16.5529154 

6.4950653 

.003649035 

275 

75625 

20796875 

16.5831240 

6.5029572 

.003636304 

276 

76176 

21024576 

16.6132477 

6.5108300 

.003623183 

277 

76729 

21253933 

16.6433170 

6.5186839 

.003610108 

278 

77284 

21484952 

16.6733320 

6.5265189 

.003597122 

279 

77841 

21717639 

16.7032931 

6.5343351 

.003584229 

280 

78400 

21952000 

16.7332005 

6.5421326 

.003571429 

281 

78961 

22188041 

16.7630546 

6.5499116 

.003558719 

282 

79524 

22425703 

16.7928556 

6.5576722 

.003546099 

283 

80089 

22G65187 

16.8226038 

6.5654144 

.003533509 

284 

80656 

22906304 

16.8522995 

6.5731385 

.003521127 

285 

81225 

23149125 

16.8819430 

6.5808443 

.003508772 

286 

81796 

23393656 

16.9115345 

6.5885323 

.003496503 

287 

82369 

23639903 

16.9410743 

6.5962023 

.003484321 

288 

82944 

23887872 

16.9705G27 

6.G038545 

.00347222-3 

289 

83521 

241375(39 

17.0000000 

6.0114890 

.003460208 

290 

84100 

24389000 

17.0293864 

6.6191060 

.003448276 

291 

84081 

24642171 

17.0587221 

6.G267054 

.003436426 

292 

85264 

24897088 

17.0880075 

6.6342874 

.003424658 

293 

85S49 

25153757 

17.1172428 

6.6418522 

.003412909 

294 

86436 

25412184 

17.1464282 

6.6493998 

.003401301 

295 

87025 

25672375 

17.1755640 

6.6569302 

.003389831 

296 

87616 

25934336 

17.2046505 

6.G644437 

.003378378 

297 

88209 

20198073 

17.2336879 

6.G719403 

.003367003 

298 

88804 

26403592 

17.2626765 

6.G794200 

.003355705 

299 

89401 

26730899 

17.2916165 

6.6868831 

.003344482 

800 

90000 

27000000 

17.3205081 

6.6943295 

.00333.3333 

301 

90601 

27270901 

17.3493510 

6.7017593 

.00:3322259 

302 

91204 

27543608 

17.3781472 

6.7091729 

.003311258 

303 

91809 

27818127 

17.4068952 

6.7165700 

.003300330 

304 

92416 

28094464 

17.4355958 

6.7239508 

.003289474 

305 

93025 

28372625 

17.4642492 

6.7313155 

.003278089 

306 

93636 

28652616 

17.4928557 

6.7386641 

.003267974 

307 

94249 

28934443 

17.5214155 

6.7459967 

.003257329 

308 

94864 

29218112 

17.5499288 

6.7533134 

.003246753 

309 

95481 

29503G29 

17  5783958 

6.7606143 

.003236246 

310 

96100 

29791000 

17!  6068169 

6.7G78995 

.003225806 

182 


TABLE  VIII.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Koots. 

Cube  Boots. 

Reciprocals. 

311 

96721 

30080231 

17.6351921 

6.7751690 

.003215434 

313 

97344 

30371328 

17.6635217 

6.7824229 

.003205128 

313 

97969 

30664297 

17.6918060 

6.7896613 

.003194888 

314 

98596 

30959144 

17.7200451 

6.7968844 

.003184713 

315 

99225 

31255875 

17.7482393 

6.8040921 

.003174603 

316 

99856 

31554496 

17.7763888 

6.8112847 

.003164557 

317 

100489 

31855013 

17.8044938 

6.8184620 

.003154574 

318 

101124 

32157432 

17.8325545 

6.8256242 

.003144654 

319 

101761 

32461759 

17.8605711 

6.8327714 

.003134796 

320 

102400 

32768000 

17.8885438 

6.8399037 

.003125000 

321 

103U41 

33076161 

17.9164729 

6.8470213 

.003115265 

322 

103684 

33386248 

17.9443584 

6.8541240 

.003105590 

323 

104329 

33698267 

17.9722008 

6.8612120 

.003095975 

324 

104976 

34012224 

18.0000000 

6.8682855 

.003086420 

325 

105625 

34328125 

18.0277564 

6.8753443 

.003076923 

326 

106276 

34645976 

18.0554701 

6.8823888 

.003067485 

327 

106929 

34965783 

18.0831413 

6.8894188 

.003058104 

328 

107584 

35287552 

18.1107703 

6.8964345 

.003048780 

329 

108241 

35011289 

18.1383571 

6.9034359 

.003039514 

330 

108900 

35937000 

18.1659021 

6.9104232 

.003030303 

331 

1005(51 

36264691 

18.1934054 

6.917'3964 

.003021148 

332 

110224 

36594368 

18.2208672 

6.9243556 

.003012048 

333 

110889 

361)26037 

18.2482876 

6.9313008 

.003003003 

334 

111556 

37259704 

18.2756669 

6.9382321 

.002994012 

335 

112225 

37595375 

18.3030052 

6.9451496 

.002985075 

336 

112896 

37933056 

18.3303028 

6.9520533 

.002976190 

337 

113569 

38272753 

18.3575598 

6.9589434 

.002967359 

338 

114244 

38614472 

18.3847763 

6.9658198 

.002958580 

339 

114921 

38958219 

18.4119526 

6.9726826 

.002949853 

340 

115600 

39304000 

18.4390889 

6.9795321 

.002941176 

341 

116281 

39651821 

18.4661853 

6.9863681 

.002932551 

342 

116964 

40001688 

18.4932420 

6.9931906 

.002923977 

343 

117649 

403:53607 

18.5202592 

7.0000000 

.002915452 

344 

118336 

40707584 

18.5472370 

7.0067962 

.002906977 

345 

119025 

41063G25 

18.5741756 

7.0135791 

.002898551 

346 

119716 

41421736 

18.6010752 

7.0203490 

.002890173 

347 

120409 

41781923 

18.6279360 

7.0271058 

.002881844 

348 

121104 

42144192 

18.6547581 

7.0338497 

.002873563 

349 

121801 

42508549 

18.6815417 

7.0405806 

.002865330 

350 

122500 

42875000 

18.7082869 

7.0472987 

.002857143 

351 

123201 

43243551 

18.7349940 

7.0540041 

.002849003 

352 

123904 

43614208 

18.7616630 

7.0006967 

.002840909 

353 

124G09 

43986977 

18.7882942 

7.0673767 

.002832861 

354 

125316 

44361864 

18.8148877 

7.0740440 

.002824859 

355 

126025 

44738875 

18.8414437 

7.0806988 

.002816901 

356 

126736 

45118016 

18.8679623 

7.0873411 

.002808989 

357 

127449 

45499293 

18.8944436 

7.0939709 

.002801120 

358 

1281  G4 

45882712 

18.9208879 

7.1005885 

.002793296 

359 

128881 

46268279 

18.9472953 

7.1071937 

.002785515 

SCO 

129600 

46656000 

18.9736660 

7.1137866 

.002777778 

361 

130321 

47045881 

19.0000000 

7.1203674 

.002770083 

362 

131044 

47437928 

19.0262976 

7.1269360 

.002762431 

363 

131769 

47832147 

19.0525589 

7.1334925 

.002754821 

364 

132496 

48228544 

19.0787840 

7.1400370 

.002747253 

365 

133225 

48627125 

19.1049732 

7.1465695 

.002739726 

366 

133956 

49027896 

19.1311265 

7.1530901 

.002732240 

367 

134689 

49430863 

19.1572441 

7.1595988 

.002724796 

368 

135424 

49836032 

19.1833261 

7.1660957 

.002717391 

369 

136161 

50243409 

19.2093727 

7.1725809 

.002710027 

370 

136900 

60653000 

19.2353841 

7.1790544 

.002702703 

37} 

137641 

51064811 

19.2613003 

7.1855162 

.002695418 

372 

138384 

51478848 

19.2873015 

7.1919663 

.002688172 

183 


TABLE  VIII.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Boots. 

Cube  Roots. 

Reciprocals. 

373 

139129 

51895117 

19.3132079 

7.1984050 

.002680965 

374 

139876 

52313G24 

19.3390796 

7.2048322 

.002673797 

375 

140625 

52734375 

19.3649167 

7.2112479 

.002666667 

376 

141376 

53157376 

19.3907194 

7.2176522 

.002659574 

377 

142129 

53582633 

19.4104878 

7.2240450 

.002652520 

378 

142884 

54010152 

19.4422221 

7.2304268 

.002645503 

379 

143641 

54439939 

19.4679223 

7.2367972 

.002638522 

380 

144400 

54872000 

19.4935887 

7.2431565 

.002631579 

381 

145161 

55306341 

19.5192213 

7.2495045 

.002624672 

382 

145924 

55742968 

19.5448203 

7.2558415 

.002617801 

383 

146689 

56181887 

19.5703858 

7.2621675 

.002610966 

884 

147456 

56623104 

19.5959179 

7.2684824 

.002604167 

385 

148225 

57066625 

19.6214169 

7.2747864 

.002597403 

386 

148996 

57512456 

19.6468827 

7.2810794 

.002590674 

387 

149769 

57960603 

19.6723156 

7.2873617 

.002583979 

388 

150544 

58411072 

19.6977156 

7.2936330 

.002577320 

389 

151321 

58863869 

19.7230829 

7.2998936 

.002570694 

390 

152100 

59319000 

19.7484177 

7.3061436 

.002564103 

391 

152881 

59776471 

19.7737199 

7.3123828 

.002557545 

392 

153664 

60236288 

19.7989899 

7.3186114 

.002551020 

393 

154449 

60698457 

19.8242276 

7.3248295 

.002544529 

394 

155.236 

61162984 

19.8494332 

7.3310369 

.002538071 

395 

156025 

61629875 

19.8746069 

7.3372339 

.002531646 

396 

156816 

62099136 

19.8997487 

7.3434205 

.002525253 

397 

157609 

62570773 

19.9248588 

7.3495966 

.002518892 

398 

158404 

63044792 

19.9499373 

7.3557624 

.002512563 

399 

159201 

63521199 

19.9749844 

7.3619178 

.002506266 

400 

160000 

64000000 

20.0000000 

7.3680630 

.002500000 

401 

160801 

64481201 

20.0249844 

7.3741979 

.002493766 

402 

161604 

64964808 

20.0499377 

7.3803227 

.002487562 

403 

162409 

65450827 

20.0748599 

7.S864373 

.002481390 

404 

163216 

65939264 

20.0997512 

7.3925418 

.002475248 

405 

164025 

66430125 

20.1246118 

f.  3986363 

.002469136 

406 

164836 

66923416 

20.1494417 

7.4047206 

.002463054 

407 

165649 

67419143 

20.1742410 

7.4107950 

.002457002 

408 

166464 

67917312 

20.1990099 

7.4168595 

.002450980 

409 

167281 

68417929 

20.2237484 

7.4229142 

.002444988 

410 

168100 

68921000 

20.2484567 

7.4289589 

.002439024 

411 

168921 

69426531 

20.2731349 

7.4349938 

.002433090 

412 

169744 

69934528 

20.2977831 

7.4410189 

.002427184 

413 

170569 

70444997 

20.3224014 

7.4470342 

.002421308 

414 

171396 

70957944 

20.3469899 

7.4530399 

.002415459 

415 

172225 

71473375 

20.3715488 

7.4590359 

.002409639 

416 

173056 

71991296 

20.3960781 

7.4650223 

.002403846 

417. 

173889 

72511713 

20.4205779 

7.4709991 

.002398082 

418 

174724 

73034632 

20.4450483 

7.4769664 

.002392344 

419 

17'5561 

73560059 

20.4694895 

7.4829242 

.002386635 

420 

176400 

74088000 

20.4939015 

7.4888724 

.002380952 

421 

177241 

74618461 

20.5182845 

7.4948113 

.002375297 

422 

178084 

75151448 

20.5426386 

7.5007406 

.002369668 

423 

178929 

75686967 

20.5669638 

7.5066607 

.002364066 

424 

179776 

76225024 

20  5912603 

7.5125715 

.002358491 

425 

180625 

76765625 

20.6155281 

7.5184730 

.002352941 

426 

181476 

77308776 

20.6397674 

7.5243652 

.002347418 

427 

182329 

77854483 

20.6639783 

7.5302482 

.002341920 

428 

183184 

78402752 

20.6881609 

7.5361221 

.002336449 

429 

184041 

78953589 

20.7123152 

7.5419867 

.002331002 

430 

184900 

79507000 

20.7364414 

7.5478423 

.002325581 

431 

185761 

800G2991 

80:7805895 

7.5536888 

.002320186 

432 

18G624 

80621568 

20.7846097 

7.5595263 

.002314815 

433 

187489 

81182737 

20.8086520 

7.5653548 

.002309469 

434 

188356 

81746504 

20.8326667 

7.5711743 

.002304147 

184 


TABLE  VIII.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Boots. 

Cube  Koots. 

Reciprocals. 

435 

189225 

82312875 

20.8566536 

7.5769849 

.002298851 

436 

190096 

82881856 

20.88061:30 

7.5827865 

.002293578 

437 

190969 

83453453 

20.9045450 

7.5885793 

.002288330 

438 

191844 

84027672 

20.9284495 

7.5943033 

.002283105 

439 

192721 

84604519 

20.9523268 

7.6001385 

.002277904 

440 

193600 

85184000 

20.9761770 

7.6059049 

.002272727 

441 

194481 

85760121 

21.0000000 

7.6116020 

.002267574 

442 

195304 

86350888 

21.0237900 

7.6174116 

.002262443 

443 

196249 

86938307 

21.0475052 

7.6231519 

.002257336 

444 

197136 

87528384 

21.0713075 

7.6288837 

.002252252 

445 

198025 

88121125 

21.0950231 

7.6346067 

.002247191 

446 

198916 

88716536 

21.1187121 

7.6403213 

.002242152 

447 

199809 

89314023 

21.1423745 

7.6460272 

.002237136 

448 

200704 

89915392 

21.1600105 

7.6517247 

.002232143 

449 

201601 

90518849 

21.1896201 

7.6574133 

.002227171 

450 

202500 

91125000 

21.2132034 

7.6630943 

.002222222 

451 

203401 

91733851 

21.2307600 

7.0687665 

.002217295 

452 

204304 

92345408 

21.2602916 

7.6744303 

.002212389 

453 

205209 

92959677 

21.2337967 

7.0800857 

.002207506 

454 

206116 

93576664 

21.3072758 

7.6857328 

.002202643 

455 

207025 

94196375 

21.3307290 

7.6913717 

.002197802 

456 

207936 

94818816 

21.a>41565 

7.6970023 

.002192982 

457 

208849 

95443993 

21.3775583 

7.7026246 

.002188184 

458 

209764 

96071912 

21.4009346 

7.7082388 

.C02183406 

459 

210081 

96702579 

21.4242853 

7.7138448 

.002178649 

460 

211600 

97336000 

21.4476106 

7.7194426 

.002173913 

461 

212521 

97972181 

21.4709106 

7.7250325 

.002169197 

402 

213444 

98611128 

21.4941853 

7.7306141 

.002164502 

463 

214369 

99252847 

21.5174348 

7.7361877 

.002159827 

464 

215296 

99897344 

21.5400592 

7.7417532 

.002155172 

465 

216225 

100544625 

21.5638587 

7.747'3109 

.002150538 

466 

217156 

101194696 

21.5870331 

7.7528606 

.002145923 

467 

218089 

101847563 

21.6101828 

7.7584023 

.002141328 

468 

219024 

102503232 

21.6333077 

7.7639361 

.002136752 

469 

219961 

103161709 

21.6564078 

7.7694620 

.002132196 

470 

220900 

103823000 

21.6794834 

7.7749801 

.002127660 

471 

221841 

104487111 

21.7025344 

7.7804904 

.032123142 

472 

222784 

105154048 

21.7255610 

7.7859928 

.C021  18644 

473 

223729 

105823817 

21.7485632 

7.7914875 

.002114165 

474 

224676 

106496424 

21.7715411 

7.7969745 

.002109705 

475 

225025 

107171875 

21.7944947 

7.80.24538 

.002105263 

476 

220576 

107850176 

21.8174242 

7.8079254 

.002100840 

477 

227529 

108531333 

21.8403297 

7.8ia3892 

.002096436 

478 

223484 

109215352 

21  8632111 

7.  C  188456 

.002092050 

479 

229441 

109902239 

21.8860086 

7.8242942 

.002087683 

480 

230400 

110592000 

21.9089023 

7.8297353 

.002088333 

481 

231301 

111284641 

21.9317122 

7.8:351088 

.002079002 

482 

232:324 

111980168 

21.9544984 

7.8405949 

.002074689 

483 

233289 

112678587 

21.9772610 

7.8460134 

.002070393 

484 

234256 

113379904 

22.0000000 

7.8514244 

.002066116 

485 

235225 

114084125 

22.0227155 

7.85682S1 

.002061856 

486 

236196 

114791256 

22.0454077 

7.8622242 

.002057613 

487 

237169 

115501303 

22.0680765 

7.8670130 

.002053388 

488 

238144 

116214272 

22.0907220 

7.8729944 

.002049180 

489 

239121 

116930169 

22.1133444 

7.8783684 

.002044990 

490 

240100 

117649000 

22.1359436 

7.8837352 

.002040816 

491 

241081 

Iia370771 

22.1585198 

7.8890916 

.002036660 

492 

242064 

119095488 

22.1810730 

7.8944-463 

.002032520 

493 

243049 

119823157 

22.2036033 

7.8997917 

.002028398 

494 

244036 

120553784 

22.2261108 

7.9051294 

.002024291 

495 

245025 

121287375 

22.24a5955 

7.9104599 

.002020202 

496  |  240016 

122023936 

22.2710575  |   7.9157832 

.002016129 

185 


TABLE  VITL-Continued. 


No. 

Squares. 

Cubes. 

Square 
Boots. 

Cube  Boots. 

Reciprocals. 

497 

'  247009 

122763473 

22.2934968 

7.9210994 

.002012072 

493 

248004 

123505992 

22.3159136 

7.9264085 

.002008032 

499 

249001 

13U51499 

22.3383079 

7.9317104 

.  003004008 

500 

250000 

125000000 

22.3606798 

7.9370053 

.002000000 

501 

251001 

125751501 

22.3830293 

7.9422931 

.001996003 

502 

252.  .04 

126506008 

22.4053565 

7.9475739 

.001992032 

503 

253009 

12?'26352r 

22.4276615 

7.9528477 

.001988072 

504 

254016 

128024064 

22.4499443 

7.9581144 

.001984127 

505 

255025 

128787625 

22.4722051 

7.9633743 

.001980198 

506 

256036 

129554216 

22.4944438 

7.9686271 

.001976285 

507 

257049 

130323843 

22.5166605 

7.9738731 

.001972387 

508 

258064 

131096512 

22.5388553 

7.9791122 

.001968504 

509 

259081 

131872229 

22.5610283 

7.9843444 

.001964037 

510 

260100 

132651000 

22.5&31796 

7.9895697 

.001900781 

511 

261121 

133432831 

22.6053091 

7.9947883 

.001950947 

513 

262144 

134217728 

22.6274170 

8.0000000 

.001953125 

513 

263169 

135005697 

22.6495033 

8.0052049 

.001949318 

514 

261196 

135796744 

22.6715681 

8.0104032 

.001945525 

515 

265225 

136590875 

22.6936114 

8.0155946 

.0019417-33 

516 

266256 

137388096 

22.7156334 

8.0207794 

.001937984 

517 

267289 

138188413 

22.73763-10 

8.0259574 

.001934236 

518 

268324 

138991832 

22.7596134 

8.0311287 

.001930502 

510 

269361 

139798359 

22.7815715 

8.0362935 

.001926782 

520 

270400 

140608000 

22.8035085 

8.0414515 

.001923077 

521 

271441 

141420761 

22.8254244 

8.04G6030 

.001919336 

522 

272484 

142236648 

22.8473193 

8.0517479 

.001915709 

523 

273529 

143055667 

22.8691933 

8.0368862 

.001912046 

524 

274576 

143877824 

22.8910463 

8.0620180 

.001908397 

525 

275625 

144703125 

22.9128785 

8.0671432 

.001904763 

526 

276676 

145531576 

22.9346899 

8.0722620 

.001901141 

527 

277729 

146363183 

22.9564806 

8.0773743 

.001897533 

528 

278784 

147197952 

22.9782506 

8.0824800 

.001893939 

529 

279841 

148035889 

23.0000000 

8.0875794 

.001890359 

530 

280900 

148877000 

23.0217289 

8.0926723 

.001886792 

531 

281961 

149721231 

23.043437'2 

8.0977589 

.001883239 

532 

283024 

150568763 

23.0051252 

8.1028390 

.001879699 

533 

284089 

151419437 

23.0867928 

8.1079128 

.001876173 

534 

285156 

152273304 

23.1084400 

8.1129803 

.001872659 

535 

286225 

153130375 

23.1300670 

8.1180414 

.001869159 

536 

2S7296 

153990656 

23.1516738 

8.1230962 

.001865672 

537 

288369 

154854153 

23.1732605 

8.1281447 

.C01862197 

538 

280444 

155720872 

23.1948270 

8.1331870 

.001858736 

539 

290521 

156590819 

23.2163735 

8.1382230 

.001855288 

540 

291000 

157464000 

23.2379001 

8.1432529 

.001851852 

541 

292681 

158340421 

23.2594067 

8.1482765 

.001848423 

542 

293764 

159220088 

23.2808935 

8.1532939 

.001845018 

543 

294S49 

16:103007 

23.3023604 

8.1583051 

.001841621 

544 

295936 

160989184 

23.3238076 

8.1633102 

.001838235 

545 

297035 

161878625 

23.3452351 

8.1683092 

.001834862 

546 

298116 

162771336 

23.3666429 

8.1733020 

.001831502 

547 

299209 

163667323 

23.3880311 

8.1782888 

.001828154 

518 

300304 

164566592 

23.4093998 

8.1832695 

.001824818 

549 

301401 

165469149 

23.4307490 

8.1882441 

.001821494 

550 

302500 

166375000 

23.4520788 

8.1932127 

.001818182 

551 

303601 

167284151 

23.4733892 

8.1981753 

.001814882 

552 

304704 

168196608 

23.4946802 

8.2031319 

.001811594 

553 

305809 

169112377 

23.5159520 

8.2080825 

.001808318 

554 

306916 

170031464 

23.5372046 

8.2130271 

.001805054 

555 

308025 

170953875 

23.5584380 

8.2179657 

.001801803 

556 

309136 

171879616 

23.5796522 

8.2228985 

.001798561 

557 

310249 

172808693 

23.600r>i74 

8.2278254 

.001795332 

558 

311364 

173741112 

83.0230336 

8.2327463      .0017:2115 

186 


TABLE  VIII.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Boots. 

Cube  Roots. 

Reciprocals. 

559 

312481 

174676879 

23.6431808 

8.2376614 

..001788909 

560 

313600 

175616000 

23.6643191 

8.2425706 

.001785714 

561 

314721 

176558481 

23.6854386 

8.247'47'40 

.001782531 

562 

315844 

177504328 

23.7065392 

8.2523715 

.001779359 

563 

316969 

178453547 

23.7'276210 

8.2572C33 

.001776199 

564 

81K096 

179406144 

23.7486842 

8.2621492 

.001773050 

565 

819225 

180362125 

23.7697286 

8.2670294 

.001769912 

566 

320356 

181321496 

23.7907545 

8.2719039 

.001766784 

567 

321489 

182284263 

23.8117618 

8.2767726 

.0017'63668 

568 

322624 

183250432 

23.8327506 

8.2816355 

.0017'60563 

509 

323761 

184220000 

#o.  6567  209 

8.J&64928 

.001757469 

170 

324900 

185193000 

23.8746728 

8.2913444 

.001754386 

571 

326041 

186169411 

23.8956063 

8.  £961903 

.001751313 

572 

327184 

187149248 

23.9165215 

8.3010304 

.001748252 

573 

3281323 

188132517 

23.  9374184 

8.3058651 

.001745201 

574 

329476 

189119224 

23.9582971 

8.3106941 

.001742160 

575 

330625 

190109375 

23.9791576 

8.3155175 

.001739130 

576 

331776 

191102976 

24.0000000 

8.  £203353 

.001736111 

577 

332929 

1921000:33 

24.0208243 

8.  £25147  5 

.001733102 

578 

334084 

1931C0552 

24.0416306 

8.8299542 

.001730104 

579 

3352-11 

1D4104529 

24.0624188 

8.3347553 

.001727116 

580 

336400 

1D5112000 

24.C831891 

8.3395509 

.001724138 

C81 

837561 

106122941 

24.1039416 

8.3443410 

.00172117'0 

582 

338724 

197137368 

24.1246762 

8.3491256 

.001718213 

583 

839889 

198155287 

24.1453929 

8.  £539047 

.001715266 

584 

341056 

199176704 

£4.1660919 

8.3586784 

.001712329 

585 

342225 

200201625 

24.1867732 

8.  £634466 

.001709402 

586 

843396 

201230056 

24.2074369 

8.3682095 

.001706485 

587 

344569 

202262003 

24.2280829 

8.87'29668 

.001703578 

588 

345744 

203297472 

£4.2487113 

8.3777188 

.001700680 

589 

346921 

204336469 

£4.£693222 

8.3824653 

.001697793 

590 

348100 

205379000 

24.2899156 

8.S872065 

.001694915 

591 

349281 

206425071 

£4.3104916 

8.3919423 

.001692047 

592 

350464 

207474688 

24.3310501 

8.  £966729 

.001689189 

593 

351649 

208527857 

24.3515913 

8.4013981 

.001686341 

594 

352836 

209584584 

24.3721152 

8.4061180 

.001683502 

595 

354025 

210644875 

24.3926218 

8.4108326 

.001680672 

596 

355216 

211708736 

24.4131112 

8.4155419 

.001677852 

597 

356409 

212776173 

24.4335834 

8.4202460 

.001675042 

598 

357604 

213847192 

24.4540385 

8.4249448 

.001672241 

599 

358801 

214921799 

24.4744765 

8.4296383 

.001669449 

600 

360000 

216000000 

24.4948974 

8.4348267 

.001666667 

C01 

361201 

£17081801 

24.5153013 

8.4390098 

.001668894 

C02 

362404 

218167208 

24.5356883 

8.4436877 

.0016611.30 

C03 

363609 

219256227 

24.5560583 

8.4483605 

.001658375 

C04 

364816 

220348864 

24.5764115 

8.4530281 

.001655629 

€05 

366025 

221445125 

24.5967478 

8.4576906 

.001652893 

606 

367236 

222545016 

24.6170673 

8.4623479 

.001650165 

607 

368449 

223648543 

24.6373,00 

8.4670001 

.001647446 

€08 

369664 

224755712 

24.6576560 

8.4716471 

.001644737 

609 

370881 

225866529 

24.6779254 

8.4762892 

.00164203li 

610 

372100 

226981000 

24.6981781 

8.4809261 

.001639344 

611 

873321 

228099131 

24.7184142 

8.4855579 

.001636661 

€12 

£74544 

229220928 

24.7386,338 

8.4901848 

.001633987 

613 

375769 

230346397 

24.7588368 

8.4948065 

.001631321 

614 

376996 

231475544 

24.7790234 

8.4994233 

.001628664 

615 

378225 

232608375 

24.7991935 

8.5040350 

.001626016 

616 

379456 

233744896 

24.8193473 

8.5086417 

.001623377 

617 

380689 

234885113 

24.8394847 

8.5132435 

.001620746 

618 

381924 

236029032     24.85960.58 

8.5178403 

.001618123 

»619 

883161    237176659     24.8797106 

8.5224321 

.001615509 

'teo 

384400    238328000  i   24.8997992 

8.5270189 

.001612903 

187 


TABLE  VIII.— Continued. 


No. 

Square 

Cubes. 

Square 
Boots. 

Cube  Roots 

Reciprocals. 

621 

385641 

239483061 

24.9198716 

8.5316009 

.001610306 

622 

386884 

240041848 

24.9399278 

8.5361780 

.001607717 

623 

388129 

241804867 

24.9599679 

8.5407501 

.001605136 

624 

389376 

242970624 

24.9799920 

8.  .5453173 

.001602564 

625 

390625 

244140625 

25.0000000 

8.5498797 

.001600000 

626 

391876 

245314376 

25.0199920 

8.:'j544372 

.001597444 

627 

393129 

246491883 

25.0399681 

8.5589899 

.001594896 

628 

394384 

247673152 

25.0599282 

8.5635377 

.001592357 

629 

395641 

248858189 

25.0798724 

8.5080807 

.001589825 

630 

396900 

250047000 

25.0998008 

8.5720189 

.001587302 

631 

398161 

251239591 

25.1197134 

8.5771523 

.001584780 

632 

399424 

252435968 

25.1396102 

8.5816809 

.001582278 

633 

400689 

253636137 

25.1594913 

8.5862047 

.001579779 

634 

401956 

254840104 

25.1793566 

8.5907238 

.001577287 

635 

403225 

256047875 

25.1992063 

8.5952380 

.001574803 

636 

404496 

257259456 

25.2190404 

8.5997476 

.001572327 

637 

405769 

258474853 

25.2388589 

8.6042525 

.001569859 

638 

407044 

259694072 

25.2586619 

8.6087526 

.001567398 

639 

408321 

200917119 

25.2784493 

8.6132480 

.001564945 

640 

409600 

262144000 

25.2982213 

8.6177388 

.001562500 

641 

410881 

263374721 

25.3179778 

8.6222248 

.001560002 

642 

412164 

264609288 

25.3377189 

8.6267063 

.001557632 

643 

413449 

265847707 

25.3574447 

8.6311830 

.001555210 

644 

414736 

267089984 

25.3771551 

8.6356551 

.001552795 

645 

416025 

268336125 

25.3968502 

8.6401226 

.001550388 

646 

417316 

269586136 

25.4165301 

8.6445855 

.001547988 

647 

418609 

270840023 

25.4361947 

8.6490437 

.001545595 

648 

419904 

272097792 

25.4558441 

8.6534974 

.001543210 

649 

421201 

273359449 

25.4754784 

8.6579465 

.001540832 

650 

422500 

274625000 

25.4950976 

8.6623911 

.001538462 

651 

423801 

275894451 

25.5147016 

8.6668310 

.001536098 

652 

425104 

277167808 

25.5342907 

8.6712665 

.001533742 

653 

426409 

278445077 

25.5538647 

8.6756974 

.001531394 

654 

427716 

279726264 

25.5734237 

8.6801237 

.001529052 

655 

429025 

281011375 

25.5929078 

8.6845456 

.001526718 

656 

430336 

282800416 

25.6124969 

8.6889630 

.001521390 

657 

431649 

283593393 

25.6320112 

8.6933759 

.001522070 

658 

432964 

284890312 

25.6515107 

8.6977843 

.001519757 

659 

434281 

286191179 

25.6709963 

8.7021882 

.001517451 

660 

435600 

287496000 

25.6904652 

8.7065877 

.001515152 

661 

436921 

288804781 

25.7099203 

8.7109827 

.001512859 

662 

438244 

290117528 

25.7293607 

8.7153734 

.001510574 

663 

439569 

291434247 

25.7487864 

8.7197596 

.001508296 

664 

440896 

292754944 

25.7681975 

8.7241414 

.001506024 

€65 

442225 

294079625 

25.7875939 

8.7285187 

.001503759 

€66 

443556 

295408296 

25.8069758 

8.7328918 

.001501502 

667 

444889 

296740963 

25.8263431 

8.7372604 

.001499250 

668 

446224 

298077632 

25.8456960 

8.741C246 

.001497006 

669 

447561 

299418309 

25.8650343 

8.7459846 

.001494708 

670 

448900 

300763000 

25.8843582 

8.7503401 

.001492537 

€71 

450241 

302111711 

25.9036677 

8.7546913 

.001490313 

€72 

451584 

303464448 

25.9229628 

8.7590383 

.001488095 

673 

452929 

304821217 

25.9422435 

8.7633809 

.001485884 

€74 

454276 

306182024 

25.9615100 

8.7677192 

.001483680 

675 

455625 

307546875 

25.9807621 

8.7720532 

.001481481 

€76 

456976 

30891*5776 

26.0000000 

8.7763830 

.001479290 

677 

458329 

310288733 

26.0192237 

8.7807084 

.001477105 

678 

459684 

311665752 

26.0384331 

8.7850296 

.001474926 

679 

461041 

313046839 

26.0576284 

8.7893466 

.004472754 

680 

462400 

314432000 

26.0768096 

8.7936593 

.001470588 

681 

463761 

315821241 

26.09597'67 

8.7979679 

.001468429 

682 

465124 

317214568 

26.1151297 

8.8022721 

.001466276 

188 


TABLE  Vm.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

683 

466489 

318611987 

26.1342687 

8.8065722 

.001464129 

684 

467856 

320013504     26.1533937 

8.8108681 

.001461988 

685 

469225 

321419125     26.1725047 

8.8151598 

.001459854 

686 

470596 

322828856 

26.1916017 

8.8194474 

.001457726 

687 

471969 

324242703 

26.2106848 

8.8237307 

.001455604 

688 

47&344 

325660672 

26.2297541 

8.8280099 

.001453488 

689 

474721 

327082769 

26.2488095 

8.8322850 

.001451379 

690 

476100 

328509000 

26.2678511 

8.8365559 

.001449275 

691 

477481 

329939371 

26.2868789 

8.8408227 

.001447178 

692 

478864 

331373888 

26.3058929 

8.8450854 

.001445087 

693 

480249 

332812557 

26.3248932 

8.8493440 

.001443001 

694 

481636 

334255384, 

26.3438797 

8.8535985 

.001440922 

695 

483025 

£35702375 

26.3628527 

8.8578489 

.001438849 

696 

484416 

337153536 

26.3818119 

8.8620952 

.001436782 

697 

485809 

338608873 

26.4007576 

8.8663375 

.001434720 

698 

487204 

340068392 

26.4196896 

8.8705757 

.001432665 

699 

488601 

341532099 

26.4386081 

8.8748099 

.001430615 

700 

490000 

343000000 

26.4575131 

8.8790400 

.001428571 

701 

491401 

344472101 

26.4764046 

8.8832661 

.001426534 

702 

492804 

345948408 

26.4952826 

8.8874882 

.001424501 

703 

494209 

347428927 

26.5141472 

8.8917063 

.001422475 

704 

495616 

348913664 

26.5329983 

8.8959204 

.001420455 

705 

497025 

350402625 

26.5518361 

8.9001304 

.001418440 

706 

498436 

351895816 

26.5706605 

8.9043366 

.001416431 

707 

499849 

353393243 

26.5894716 

8.9085387 

.001414427 

708 

501264 

354894912 

26.6082694 

8.9127369 

.001412429 

709 

502681 

356400829 

26.6270539 

8.9169311 

.001410437 

'710 

504100 

357911000 

26.6458252 

8.9211214 

.001408451 

711 

505521 

359425431 

26.6645833 

8.9253078 

.001406470 

712 

506944 

360944128 

26.6833281 

8.9294902 

.001404494 

713 

508369 

362467097 

26.7020598 

8.9336687 

.001402525 

714 

509796 

363994344 

26.7207784 

8.9378433 

.001400560 

715 

511225 

365525875 

26.7394839 

8.9420140 

.001398601 

716 

512656 

367061696 

26.7581763 

8.9461809 

.001396648 

717 

514089 

368601813 

26.7768557 

8.9503438 

.001394700 

718 

515524 

370146232 

26.7955220 

8.9545029 

.001392758 

719 

516961 

371694959 

26.8141754 

8.9586581 

.001390821 

720 

518400 

373248000 

26.8328157 

8.9628095 

.001388889 

721 

519841 

374805361 

26.8514432 

8.9639570 

.(,01386963 

722 

521284 

376367048 

26.8700577 

8.9711007 

.001385042 

723 

522729 

377933067 

26.8886593 

8.9752406 

.001383126 

724 

524176 

379503424 

26.9072481 

8.9793766 

.001381215 

725 

525625 

381078125 

26.9258240 

8.9835089 

.001379310 

726 

527076 

382657176 

26.9443872 

8.9876373 

.001377410 

727 

528529 

384240583 

26.9629375 

8.9917620 

001375516 

728 

529984 

385828352 

26.9814751 

8.9958829 

.001373626 

729 

531441 

387420489 

27.0000000 

9.0000000 

.001371742 

730 

532900 

389017000 

27.0185122 

9.0041134 

.001369863 

731 

534361 

390617891 

27.0370117 

9.0082229 

.001367989 

732 

535824 

392223168 

27.0554985 

9.0123288 

.001366120 

733 

537289 

393832837 

27.0739727 

9.0164309 

.001364256 

734 

538756 

395446904 

27.0924344 

9.0205293 

.001362:398 

735 

540225 

397065375 

27.1108834 

9.0246239 

.001360544 

736 

541696 

398688256 

27.1293199 

9.0287149 

.00ia58696 

737 

543169 

400315553 

27.1477439 

9.0328021 

.001356852 

738 

544644 

401947272 

27.1661554 

9.0368857 

.001355014 

739 

546121 

403583419 

27.1845544 

9.0409G55 

.001353180 

740 

547600 

405224000 

27.2029410 

9.0450419 

.001351351 

741 

549081 

406869021 

27.2213152 

9.0491142 

.001349528 

742 

550564 

408518488 

27.2396769 

9.0531831 

.001347709 

,743 

552049 

410172407 

27.2580263 

9.0572482 

.001345895 

744 

553536 

411*30784  |   27.2763634 

9.0613098 

.001344086 

TABLE  VIIL— Continued. 


''NO. 

Squares. 

Cubes. 

Square 
Boots. 

Cube  Roots. 

Reciprocals. 

745 

555025 

413493625 

27.2946881 

9.0653677 

.0013:2282 

746 

556516 

415160936 

27.3130006 

9.0094220 

.001340483 

747 

558009 

4168327'23 

27.3313007 

9.07'34726 

.001338088 

748 

559504 

4185081)92 

27.3495887 

.9.0775197 

.001336898 

749 

561001 

420189749 

27.3678044 

5).  0815031 

.001335113 

750 

562500 

421875000 

27.3861379 

9.085G030 

.001333333 

751 

5(54001 

428564751 

27.4043r<U2 

9.  0890392 

.001331558 

752 

565504 

425259008 

27.4220184 

9.0930719 

,00132978? 

753 

567009 

426957777 

27.4408455 

9.0977010 

.001328021 

754 

568516 

428001004 

27.4590004 

9.1017205 

.001320200 

755 

570025 

430308875 

27.4772033 

9.1057'485 

.001324503 

756 

571536 

432081216 

27.4954542 

9.1(/J7'G69 

.CC1122751 

757 

573049 

483798093 

27.5130330' 

9.11^7818 

.GG1321004 

758 

574564 

435519512 

27.5317998 

9.1177931 

.001319201 

759 

576081 

437245479 

27.5499546 

9.1218010 

.G01i>17523 

760 

577600 

438976000 

27.5680975 

9.1258053 

.001815789 

761 

579121 

440711081 

27.5862284 

9.1X96001 

.101314000 

762 

580644 

442450728 

27.604347'5 

9.13o8034 

.001312336 

763 

582169 

444194947 

27.62^4540 

9.1377971 

.001310010 

764 

583696 

445943744 

27.6405499 

9.1417874 

.001308901 

765 

585225 

447'697125 

27.0586334 

9.1457742 

.001  307190 

766 

586756 

449455096 

27.6767050 

9.1497576 

.001305483 

767 

588289 

451217663 

27.0947'G48 

9.1537'375 

.001:^03781 

768 

589824 

452984832 

27.7128129 

<J.15771S9 

.001308088 

769 

591361 

454750009 

27.7308492 

(J.101G80'J 

.001300390 

770 

592900 

456533000 

27.7488739 

9.1656565 

.001298701 

771 

594441 

458314011 

27.7008868 

9.1096225 

.001*97017 

772 

595984 

460099048 

27.7848880 

9.1735852 

.001*95337' 

773 

597529 

461889917 

27.8028770 

9.1775445 

.001293001 

'774 

599076 

463684824 

27.8208555 

9.1815003 

.001291990 

775 

600625 

465484375 

27.8388218 

9.1854527 

.0012U0323 

776 

602176 

467288570 

27.b567760 

9.1b94018 

.001288060 

777 

603729 

469097433 

27.8747197 

9.1933474 

.001287001 

778 

605284 

470910952 

27.8926514 

9.1972897 

.001*85847 

779 

606841 

472729139 

27.9105715 

9.2012286 

.001283097 

780 

608400 

474552000 

27.9284801 

9.2051641 

.001282051 

781 

609961 

476379541 

27.9463772 

9.  2010962 

.001280410 

782 

611524 

478211708 

27.9642029 

9.*lcO*50 

.001278772 

783 

613089 

480048687 

27.9821372 

9.2109505 

.001*77139 

784 

614656 

481890304 

28.0000000 

9.2208720 

.001275510 

785 

616225 

483730025 

28.0178515 

9.2247914 

.001*7U885 

786 

617796 

485587056 

28.0356915 

9.2267008 

.001272205 

787 

619369 

4874434C3 

28.0535203 

9.23*Glb9 

.001*70048 

788 

620944 

489303872 

28.0713377 

9.2305277 

.G012WJ030 

789 

622521 

491109009 

28.0891438 

9.2404333 

.001*07427 

790 

624100 

493039000 

28.1069386 

9.2443355 

.001205823 

791 

625681 

494913071 

28.1247'222 

9.2482344 

.001204223 

792 

627264 

496793088 

28.1424946 

9.2521300 

.001202620 

793 

628849 

498677257 

28.1602557 

9.2500224 

.001201034 

794 

630436 

500566184 

28.1780056 

9.2599114 

.001*59440 

795 

632025 

50245987'5 

28.1957444 

9.2037973 

.001257862 

796 

633616 

504358336 

28.2134720 

9.2076798 

.001250281 

797 

635209 

506201573 

28.2311884 

9.2715592 

.0015:54705 

798 

636804 

508169592 

28.2488938 

9.2754352 

.001253133 

799 

638401 

510082399 

28.2665881 

9.2793081 

.001251504 

800 

640000 

512000000 

28.2842712 

9.2831777 

.001250000 

801 

641601 

513922401 

28.8019434 

9.2870440 

.001248439 

802 

643204 

515849608 

28.3196045 

9.2909072 

.001246883 

803 

644809 

517781627 

28.3372546 

9.2947071 

.001245330 

804 

646416 

5197184&4 

28.3548938 

9.2986239 

.001243781 

805 

648025 

521660125 

28.3725219 

9.3024775 

.001242236 

806 

649636 

523606616 

28.3901391 

9.3003278 

.001240695 

190 


TABLE   VIII.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Boots. 

Cube  Roots. 

Reciprocals. 

807 

6512-19 

525557943 

28.4077454 

9.3101750 

.001239157 

803 

652864 

527514112 

28.4253408 

9.3140190 

.001237624 

809 

654481 

529475129 

28.4429253 

9.3178599 

.001236094 

810 

656100 

531441000 

28.4604989 

9.3216975 

.001234568 

811 

657721 

533411731 

28.4780617 

9.3255320 

.001233046 

812 

659344 

535387328 

28.4956137 

9.3293634 

.001231527 

813 

660969 

537367797 

28.5131549 

9'.  3331816 

.001230012 

814 

662596 

539353144 

28.5306852 

9.3370167 

.001228501 

815 

664225 

541343375 

28.5482048 

9.3408386 

.001226994 

816 

665856 

543338496 

28.5657137 

9.3446575 

.001225490 

817 

667489 

545338513 

28.5832119 

9.3484731 

.001223990 

818 

669124 

547343432 

28.6006993 

a.  3522857 

.001222494 

819 

670761 

549353259 

28.6181760 

9.3560952 

.001221001 

820 

672400 

551368000 

28.6356421 

9.3599016 

.001219512 

821 

674041 

553:387661 

28.6530976 

9.3637'049 

.001218027 

822 

675684 

555412248 

28.6705424 

9.3675051 

.001216545 

823 

677329 

557441767 

28.  68797  G6 

9.3713022 

.001215067 

824 

678976 

559476224 

28.7054002 

9.3750963 

.001213592 

825 

680625 

561515625 

28.7228132 

9.3788873 

.001212121 

828 

682276 

563559976 

28.7402157 

9.3826752 

.001210654 

837 

683929 

565609283 

28.7576077 

9.3864600 

.001209190 

838 

685584 

567663552 

28.7749891 

9.3902419 

.001207729 

829 

687241 

569722789 

28.7923601 

9.3940206 

.001206273 

830 

688900 

571787000 

28.8097206 

9.3977964 

.001204819 

831 

690561 

573856191 

28.82707'06 

9.4015691 

.001203369 

832 

692224 

575930368 

28.8444102 

9:4053387 

.001201923 

8-33 

693889 

578009537 

28.8617394 

9.4091054 

.001200480 

834 

695556 

580093704 

28.8790582 

9.4128690 

.001199041 

835 

697225 

582182875 

28.8963666 

9.4166297 

.001197605 

836 

698890 

584277056 

28.9136646 

9.4203873 

.001196172 

637 

700569 

586376253 

28.9309523 

9.4241420 

.001194743  "" 

833 

702244 

588480472 

28.9482237 

9.4278936 

.001193317 

839 

703921 

590589719 

28.9654967 

9.4316423 

.001191895 

840 

705600 

592704000 

28.9827535 

9.4353880 

.001190470 

841 

707281 

59482:3321 

29.0000000 

9.4391307 

.001189061 

842 

708964 

596947688 

29.0172363 

9.4428704 

.001187648 

843 

710649 

599077107 

29.0344023 

9.4466072 

.001186240 

844 

712336 

601211584 

29.05167'81 

9.450ailO 

.001184834 

845 

714025 

603351125 

29.0688837 

9.4540719 

.001183432 

846 

715716 

605495736 

29.08607D1 

9.4577999 

.001182033 

847 

717409 

607645423 

29.1032644 

9.4615249 

.001180633 

848 

719104 

609800192 

29.1204396 

9.4652470 

.001179245 

849 

720801 

611960049 

29.1376046 

9.4689661 

.001177856 

850 

722500 

614125000 

29.1547595 

9.4726824 

.001176471 

851 

724201 

616295051 

29.1719043 

9.4763957 

.001175088 

852 

725904 

618470208 

29.1890390 

9.4801061 

.001173709 

853 

727609 

620650477 

29.2061637 

9.4838136 

.001172333 

854 

729316 

622835864 

29.2232784 

9.4875182 

.001170960 

855 

731025 

625026375 

29.2403830 

9.4912200 

.001169591 

853 

732736 

627222016 

29.2574777 

9.4949188 

.001168224 

857 

734449 

629422793 

29.2745623 

9.4986147 

.001166861 

858 

73()1G4 

631628712 

29.2916370 

9.5023078 

.001165501 

859 

737881 

633839719 

29.3087018 

9.5059980 

.001164144 

860 

739600 

636056000 

29.3257566 

9.5096854 

.001162791 

861 

741321 

638277381 

29.3428015 

9.5133699 

.001161440 

862 

743044 

640503928 

29.  £598365 

9.5170515 

.001160093 

863 

744769 

642735647 

29.3768616 

9.5207303 

.001158749 

864 

746496 

644972544 

29.3938769 

9.5244063 

.001157407 

865 

748225 

64721  4625 

29.4108823 

9.5280794 

.001156069 

866 

749956 

649461896 

29.4278779 

9.5317407 

.001154734 

867 

751689 

651714363 

29.4448ffl7 

9.5354172 

.001153403 

£68 

753434 

653972032    29.4618397     9.5390818     001152074 

191 


TABLE  V1IL.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Hoots. 

Cube  Roots. 

Reciprocals. 

869 

755161 

656234909 

29.4788059 

9.5427437 

.001150748 

870 

756900 

658503000 

29.4957624 

9.5464027 

.001149425 

871 

758641 

660776311 

29.5127091 

9.5500589 

.001148106 

872 

760384 

663054848 

29.5296461 

9.5587123 

.001146789 

873 

762129 

665338617 

»  .5466784 

9.5573030 

.001145475 

874 

763876 

667627624 

29.5634910 

9.5610108 

.001144165 

875 

765625 

669921875 

29.5803989 

9.5646559 

.001142857 

876 

767376 

67'2221376 

29.5972972 

9.5682982 

.001141553 

877 

769129 

674526133 

29.6141858 

9.5719377 

.001140251 

878 

770884 

676836152 

29.6310648 

9.5755745 

.001138952 

879 

772641 

679151439 

29.6479342 

9.5792085 

.001137656 

880 

774400 

681472000 

29.6647939 

9.5828397 

.001136364 

881 

776161 

683797841 

29.6816442 

9.5864682 

.001135074 

882 

777924 

686128968 

29.6984848 

9.5900939 

.001133787 

883 

779689 

688465387 

29.7153159 

9.5937169 

.001132503 

884 

781456 

690807104 

29.7321375 

9.5973373 

.001131222 

885 

783225 

693154125 

29.7489496 

9.6009548 

.001129944 

886 

784996 

695506456 

29.7657'521 

9.6045096 

.001128668 

887 

786769 

697864103 

29.7825452 

9.0081817 

.001127396 

888 

788544 

700227072 

29.7993289 

9.6117911 

.001126126 

889 

790321 

702595369 

29.8101030 

9.G153977 

.001124859 

890 

792100 

704989000 

29.8328678 

9.6190017 

.001123596 

891 

793881 

707347971 

£9.  8496231 

9.0220030 

.001122334 

892 

795664 

709732288 

29.8663G90 

9.6202016 

.001121076 

893 

797449 

712121957 

29.8831056 

9.6297975 

.001119821 

894 

799236 

714516984 

29.8998328 

9.0333907 

.001118568 

895 

801025 

716917375 

29.9165506 

9.G369812 

.001117318 

896 

802816 

719323136 

29.9332591 

9.6405090 

.001116071 

897 

804609 

721734273 

29.9499583 

9.6441542 

.001114827 

898 

806404 

724150792 

29.9666481 

9.6477367 

.001113586 

.899 

808201 

726572699 

29.9833287 

9.6513166 

.001112347, 

900 

810000 

729000000 

30.0000000 

9.6548938 

.001111111 

901 

811801 

731432701 

30.0166620 

9.0584084 

.001109873 

902 

813604 

733870808 

30.0333148 

9.GG20403 

.001108647 

903 

815409 

736314327 

30.0499584 

9.6650096 

.001107420 

904 

817'216 

738763264 

30.06G5928 

9.6691762 

.0011C6195 

905 

819025 

741217625 

30.0832179 

9.6727403 

.001104972 

906 

820836 

743677416 

30.0998339 

9.6763017 

.001108753 

907 

822649 

746142643 

30.1164407 

9.6798604 

.001102^3(5 

908 

824464 

748613312 

30.1330383 

9.6834166 

.001101322 

909 

826281 

751089429 

30.1496269 

9.6869701 

.001100110 

910 

828100 

753571000 

30.1662063 

9.0905211 

.001098901 

911 

829921 

756058031 

30.1827765 

9.0940094 

.001097095 

912 

831744 

758550528 

30.1993377 

9.6976151 

.001096491 

913 

833569 

761048497 

30.2158899 

9.7011583 

.001095290 

914 

835396 

763551944 

30.2324329 

9.7046989 

.001094002 

915 

837225 

766060875 

30.2489669 

9.7082369 

.001092896 

916 

839056 

768575296 

30.2654919 

9.7117723 

.001091703 

917 

840889 

771095213 

30.2820079 

9.7153051 

.001090513 

918 

842724 

773620632 

30.2985148 

9.7188354 

.001089335 

919 

844561 

776151559 

30.3150128 

9.7223631 

.001088139 

920 

846400 

778688000 

30.&315018 

9.7258883 

.001086957 

921 

848241 

781229961 

30.3479818 

9.7294109 

.00108577'6 

922 

850084 

783777448 

30.3644529 

9.7329309 

.001084599 

923 

851929 

786330467 

30.3809151 

9.7304484 

.001083423 

924 

853776 

788889024 

30.3973683 

9.7399634 

.001082251 

925 

855625 

791453125 

30.4138127 

9.7434758 

.001081081 

926 

857476 

794022776 

30.4302481 

9.7469857 

.001079914 

927 

859329 

796597983 

30.4466747 

9.7504930 

.001078749 

928 

861184 

799178752 

30.4630924 

9.75S9979 

.001077580 

9S9 

863041 

801765089 

30.4795013 

9.7575002 

.001076426 

930 

864900 

804357000 

30.4959014     9.7610001 

.001075269 

TABLE  VTH.— Continued. 


No. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots,  j  Reciprocals. 

931 

866761  j   806954491 

30.5122926 

9.7644974 

.001074114 

932 

868624  J   809557568 

30.52S6750 

9.7679922 

.001072961 

933 

870489 

812166237 

30.5450487 

9.7714845 

.001071811 

934 

872356 

814780504 

30.5614136 

9.77'49743 

.001070664 

935 

874225 

817400375 

30.5777697 

9.7784616 

.001069519 

936 

876096 

820025856 

30.5941171 

9.7819466 

.001068376 

937 

877969 

822656953 

30.6104557 

9.7854288 

.001067236 

938 

§79844 

825293672 

30.6267857 

9.7889087 

.001066098 

939 

881721 

827936019 

30.6431069 

9.7923861 

.001064963 

940 

883600 

830584000 

30.6594194 

9.7958611 

.001063830 

941 

885481 

833237621 

30.6757233 

9.7993336 

.001062699 

942 

887364 

835896888 

30.6920185 

9.8028036 

.001061571 

943 

889249 

838561807 

30.7083051 

9.8062711 

.001060445 

944 

891136 

841232384 

:.0.  7245830 

9.8097362 

.001059322 

945 

893025 

843908625 

30.7408523 

9.8131989 

.001058201 

946 

894916 

846590536 

30.7571130 

9.8166591 

.001057082 

947 

896809 

849278123 

30.7733651 

9.8201169 

.001055966 

948 

898704 

851971392 

30.78C6G86 

9.8235723 

.001054852 

949 

900601 

854670349 

30.8058436 

9.8270252 

.001053741 

950 

902500 

857375000 

30.8220700 

9.8304757 

001052632 

951 

904401 

860085351 

30.8382879 

9.8338238 

.001051525 

952 

906304 

862801408 

30.8544972 

9.8373695 

.001050420 

953 

908209 

865523177 

30.8706981 

9.8408127 

.001049318 

954 

910116 

868250664 

30.8868904 

9.8442536 

.001048218 

955 

912025 

870983875 

30.9030743 

9.8476920 

.001047120 

956 

913936 

873722816 

30.9192497 

9.8511280 

.001046025 

957 

915849 

876467493 

30.9354166 

9.8545617 

.001044932 

958 

917764 

879217912 

30.9515751 

9.8579929 

.001043841 

959 

919681 

881974079 

30.9677251 

9.&614218 

.001042753,, 

,960 

921600 

884736000 

SO.  9838668 

9.8648483 

.00104166A 

'061 

923521 

887503681 

31.0000000 

9.8682724 

.001040583 

962 

925444 

£90277128 

31.0161248 

9.8716941 

.001039501 

063 

927369 

893056347 

31.0322413 

9.87'51135 

.001038422 

964 

929296 

£95841344 

31.0483494 

9.8785305 

.001037344 

965 

931225 

198632125 

31.0644491 

9.8819451 

.001036269 

966 

933156 

901428696 

31.0805405 

9.885357'4 

.001035197 

967 

935089 

904231063 

31.0966236 

9.8887673 

.001134126 

968 

937024 

007039232 

31.1126984 

9.8921749 

.00103EC58 

969 

938961 

909853209 

31.1287648 

9.8955801 

.(J01031092 

070 

940900 

912673000 

31.1448230 

9.8989830 

.C0103C928 

971 

942841 

915498611 

31.16C6729 

9.9C28835 

.00102C8G6 

972 

944784 

918330048 

31.1769145 

9.0057817 

.001C2fc807 

973 

946729 

921167317 

31.1929479 

9.9091776 

.001027749 

974 

948676 

924010424 

31.2089731 

9.9125712 

.C01  026694 

975 

950625 

926859375 

31.2249900 

9.9150624 

.C01025641 

976 

952576 

929714176 

31.2409987 

9.9108513 

.001024590 

977 

954529 

932574833 

31.2569992 

9.0827379 

.001023541 

978 

956484 

935441352 

31.2729915 

9.0261222 

.001022495 

979 

958441 

038313739 

31.2889757 

9.0295042 

.001021450 

980 

960400 

941192000 

31.3049517 

9.9328839 

.001020408 

981 

'962361 

944076141 

31.3209195 

9.9362613 

.001010368 

082 

964324 

946966168 

31.3368792 

9.9396363 

.001018330 

9a3 

966289 

9498G2087 

31.3528308 

9.9430092 

.001017294 

984 

968256 

952763904 

31.3687743 

9.9463797 

.001016260 

985 

970225 

955671  C25 

31.3847097 

9.9497479 

.001015228 

986 

972196 

958585256 

31.4006369 

9.9531138 

.001014199 

987 

974169 

961504803 

31.4165561 

9.9564775 

.001013171 

988 

976144 

964430272 

31.4324673 

9.9598389 

.001012146 

989 

978121 

967361669 

31.4483704 

9.9631981 

.001011122 

990 

980100 

970299000 

31.4642654 

9.9665549 

.001010101 

\801 

982081 

973242271 

31.4801525 

9.9699095 

.001009082f' 

992 

984064 

976191488 

31.4960315 

9.9732619 

.001008066 

193 


TABLE  VIII.— Continued. 


>. 

Squares. 

Cubes. 

Square 
Roots. 

Cube  Roots. 

Reciprocals. 

993 

986049 

979146657 

31.5119025 

9.9766120 

.001007049 

994 

988036 

982107784 

31.5277655 

9.9799599 

.001006036 

995  . 

990025 

085074875 

31.5436206 

9.9833055 

.001005025 

996 

992010 

988047936 

31.5594677 

9.9866488 

.001004016 

997 

994003 

991026973 

31.5753068 

9.9899900 

.001003009 

993 

99G004 

994011992 

31.5911380 

9.9933289 

.001002004 

999 

998301 

997002999 

31.6069613 

9.99GG056 

.  .001001001 

1000 

1000000 

1000000000 

31.6227766 

10.0000000 

.001000000 

1001 

1002001 

1003003001 

31.6385840 

10.0033322 

.0009990010 

100  2 

1034004 

1006012008 

31.6543836 

10.0066622 

.0009980040 

1003 

1006009 

1009027027 

31.6701752 

10.0099899 

.0009970090 

1004 

1008016 

1012J48064 

31.6859590 

10.0133155 

.0009960159 

1005 

1010025 

1015075125 

31.7017349 

10.0166389 

.0009950249 

1008 

1012036 

1018108216 

31.7175030 

10.0199601 

.0009940358 

1007 

1014049 

1021147'343 

31.7332633 

10.0232791 

.0009930487 

1003 

1016064 

1024192512    31.7490157 

10.0265958 

.0009920635 

1009 

1018081 

1027243729    31.7647603 

10.0299104 

.0009910803 

1010 

1020100 

1030301000  j  31.7804972 

10.0332228 

.0009900990 

1011 

1022121 

1033364331  ]  31.79622G2 

10.0365330 

.0009891197 

1012 

1024144 

103S433723 

31.8119474 

10.0398410 

.0009881423 

1013 

1026169 

1033509197 

31.8276609 

10.0431469 

.0009871668 

1014 

1028196 

1042593744 

31.8433666 

10.0464506 

.0009861933 

1015 

1030225 

1045678375 

31.8590646 

10.0497521 

.0009852217 

1016 

103-2256 

1048772096 

31.8747549 

10.0530514 

.0009842520 

ioir 

1034289 

1051871913 

31.8904374 

10.0563485 

.0009832842 

1018 

103G324 

1054977832 

31.9061123 

10.0596435 

.0009823183 

1019 

1038361 

1053089859 

31.9217794 

10.0629364 

.0009813543 

.1020 

1040400 

1061208000 

31.9374388 

10.0662271 

.0009803922 

\1021 

1042441 

1064332261 

31.9530906 

10.0695156 

.0009794319 

1022 

1044484 

1067462648 

31.9637347 

10.0728020 

.0009784736 

1023 

1046529 

1070599167 

31.9843712 

10.0760863 

.0009775171 

1024 

1048576 

1073741824 

32.1.000000 

10.0793G84 

.0009765625 

1025 

1050625 

1076890625 

32.0156212 

10.0826484 

.0009756098 

1026 

1052676 

1080045576 

32.0312348 

10.0859262 

.0009746589 

1027 

1054729 

1083206683 

32.0468407 

10.0892019 

.0009737098 

1028 

1056784 

1086373952 

32.0624391 

10.0924755 

.0009727626 

1029 

1058841 

1039547389 

32.0780298 

10.095746D 

.0009718173 

1030 

1060900 

1092727000 

32.0936131 

10.0990163 

.0009708733 

1031 

1062961 

1095912791 

32.1091887 

10.1022835 

.0009699321 

1032 

1065024 

1099104768 

32.1247568 

10.1055487 

.00  9689922 

1033 

1067089 

1102302937 

32.1403173 

10.1088117 

.0009680542 

1034 

1069156 

1105507304 

32.1558704 

10.1120726 

.0009671180 

1035 

1071225 

1108717875 

32.1714159 

10.1153314 

.0009661836 

1036 

1073296 

1111934656 

32.1869539 

10.1185882 

.0009652510 

1037 

1075369 

1115157653 

32.2024844 

10.1218428 

.0009643202 

1038 

1077444 

1118386872 

32.2180074 

10.1250953 

.0009633911 

1039 

1079521 

1121622319 

32.2335229 

10.1283457 

.0009624639 

1040 

1081600 

1124864000 

32.2490310 

10.1315941 

.0009615385 

1041 

1083681 

1128111921 

32.2645316 

10.1348403 

.0009606148 

1042 

1085764 

1131366088 

32.2800248 

10.1380845 

.  .0009596929 

1043 

1087849 

1134626507 

32.2955105 

10.1413266 

.0009587733 

1044 

1089936 

1137893184 

32.3109888 

10.1445667 

.0009578544 

1045 

1092025 

1141166125 

32.3264598 

10.1478047 

.0009569378 

1046 

1094116 

1144445336 

32.3419233 

10.1510406 

.0009560229 

1047 

1096209 

1147730823 

32.a573794 

10.1542744 

.0009551098 

1048 

1098304 

1151022592 

32.3728281 

10.1575062 

.0009541985 

1049 

1100401 

1154320649 

32.3882695 

10.1607359 

.0009532888 

1050 

1102500 

1157625000 

32.4037035 

10.1639636 

.0009523810 

1051 

1104601 

1160935651 

32.4191301 

10.1671893 

.0009514748 

1052 

1106704 

1164252608 

32.4345495 

10.1704129 

-  .0009505703 

Viass 

1108809 

1167575877 

32.4499615 

10.1736344 

.0009496676 

1054 

1110916 

1170905464 

32.4653662 

10.1768539 

.0009487666 

194 


TABLE   IX.  —  LOGARITHMS   OF   NUMBERS. 


NO123456789 


100 

1 
2 
3 
4 
5 
6 
7 
8 
9 


4 
5 
6 

7 
8 
9 

120 

1 
2 
3 
4 
5 
6 
7 
8 
9 

130 

1 
2 
3 
4 
5 
6 
7 
8 
9 

140 

1 
2 
3 

4 


00000  00043  00087  00130  00173  00217  00260  00303  00346  00389 

0432  0475  0518  0561  0604  0647  0689  0732  0775  0817 

0860  0903  0945  0988  1030  1072  1115  1157  1199  1242 

1284  1326  1368  1410  1452  1494  1536  1578  1620  1662 

1703  1745  1787  1828  1870  1912  1953  1995  2036  2078 

2119  2160  2202  2243  2284  2325  2366  2407  2449  2490 

2531  2572  2612  2653  2694  2735  2776  2816  2857  2898 

2938  2979  3019  3060  3100  3141  31»1  3222  3262  3302 

3342  3383  3423  3463  3503  3543  3583  3623  3663  3703 

3743  3782  3822  3862  3902  3941  3981  4021  4060  4100 

04139  04179  04218  04258  04297  04336  04376  04415  04454  04493 

4532  4571  4610  4650  4689  4727  4766  4805  4844  4883 

4922  4961  4999  5038  5077  5115  5154  5192  5231  5269 

5308  5346  5385  5423  5461  5500  5538  5576  5614  5(552 

5690  5729  5767  5805  5843  5881  5918  5956  5994  6032 

6070  6108  6145  6183  6221  6258  6296  6333  6371  6408 

6446  6483  6521  6558  6595  6633  6670  6707  6744  6781 

6819  6856  6893  6930  6967  7004  7041  7078  7115  7151 

7188  7225  7262  7298  7335  7372  7408  7445  7482  7518 

7555  7591  7628  7664  7700  7737  7773  7809  7846  7882 

07918  07954  07990  08027  08063  08099  08135  08171  08207  08243 

8279  8314  8350  8386  8422  8458  8493  8529  8565  8600 

8636  8672  8707  8743  8778  8814  8849  8884  8920  8955 

8991  9026  9061  9096  9132  9167  9202  9237  9272  9307 

9342  9377  9412  9447  9482  9517  9552  9587  9621  9656 

9691  9726  9760  9795  9830  9864  9899  9934  9968  10003 

10037  10072  10106  10140  10175  10209  10243  10278  10312  0346 

0380  0415  0449  0483  0517  0551  0585  0619  0653  0687 

0721  0755  0789  0823  0857  0890  0924  0958  0992  1025 

1059  1093  1126  1160  1193  1227  1261  1294  1327  1361 

11394  11428  11461  11494  11528  11561  11594  11628  11661  11694 

1727  1760  1793  1826  1860  1893  1926  1959  1992  2024 

2057  2090  2123  2156  2189  2222  2254  2287  2320  2352 

2385  2418  2450  2483  2516  2548  2581  2613  2646  2678 

2710  2743  2775  2808  2840..  2872  2905  2937  2969  3001 

3033  3066  3098  3130  3162  3194  3226  3258  3290  3322 

3354  3386  3418  3450  3481  3513  3545  3577  3609  3640 

3672  3704  3735  3767  3799  3830  3862  3893  3925  3956 

3988  4019  4051  4082  4114  4145  4176  4208  4239  4270 

4301  4333  4364  4395  4426  4457  4489  4520  4551  4582 

14613  14644  14675  14706  14737  14768  14799  14829  14860  14891 

4922  4953  4983  5014  5045  5076  5106  5137  5168  5198 

5229  5259  5290  5320  5351  5381  5412  5442  5473  5503 

5534  5564  5594  5625  5655  5685  5715  5746  5776  5806 

5836  5806  5897  5927  5957  5987  6017  6047  6077  6107 


5  6137  6167  6197  6227^  6256  6286  6316  6346  6376  6406 

6  6435  6465  6495  6524  6554  6584  6613  6643  6673  6702 

7  6732  6761  6791  6820  6850  6879  6909  6938  6967  6997 

8  7026  7056  7085  7114  7143  7173  7202  7231  7260  7289 

9  7319  7348  7377  7406  7435  7464  7493  7522  7551  7580 

>0  !  17609  17638  17667  17696  17725  17754  17782  17811  17840  17869 


TABLE   IX.  —  LOGARITHMS   OF   NUMBERS. 
NO1234567    ~~8 


150  17609  17638  17667  17696  17725  17754  17782  17811  17840  17869 

1  7898  7926  7955  7984  8013  8041  8070  8099  8127  8156 

2  8184  8213  8241  8270  8298  8327  8355  8384  8412  8441 

3  8469  8498  8526  8554  8583  8611  8639  8667  8696  8724 

4  8752  8780  8808  8837  8865  8893  8921  8949  8977  9005 

5  9033  9061  9089  9117  9145  9173  9201  9229  9257  9285 

6  9312  9340  9368  9396  9424  9451  9479  9507  9535  9562 

7  9590  9618  9(545  9673  9700  9728  9756  9783  9811  9838 

8  9866  9893  9921  9948  99762000320030200582008520112 

9  20140  20167  20194  20222  20249  0276  0303  0330  0358  0385 

160  20412  20439  20466  20493  20520  20548  20575  20602  20629  20656 

1  0683  0710  0737  0763  0790  0817  0844  0871  0898  0925 

2  0952  0978  1005  1032  1059  1085  1112  1139  1165  1192 

3  1219  1245  1272  1299  1325  1352  1378  1405  1431  1458 

4  1484  1511  1537  1564  1590  1617  1643  1669  1696  1722 

5  1748  1775  1801  1827  1854  1880  1906  1932  1958  1985 

6  2011  2037  2063  2089  2115  2141  2167  2194  2220  2246 

7  2272  2298  2324  2350  2376  2401  2427  2453  2479  2505 

8  2531  2557  2583  2608  2634  2660  2686  2712  2737  2763 

9  2789  2814  2840  2866  2891  2917  2943  2968  2994  3019 

170  23045  2307JO  23096  23121  23147  23172  23198  23223  23249  23274 

1  3300  3325  3350  3376  3401  3426  3452  3477  3502  3528 

2  3553  3578  3603  3629  3654  3679  3704  3729  3754  3779 

3  3805  3830  3855  3880  3905  3930  3955  3980  4005  4030 

4  4055  4080  4105  4130  4155  4180  4204  4229  4254  4279 

5  4304  4329  4353  4378,4403  4428  4452  4477  4502  4527 

6  4551  4576  4601  4625  4650  4674  4699  4724  4748  4773 

7  4797  4822  4846  4871  4895  4920  4944  4969  4993  5018 

8  5042  5066  5091  5115  5139  5164  5188  5212  5237  5261 

9  5285  5310  5334  5358  5382  5406  5431  5455  5479  5503 

180  25527  25551  25575  25600  25624  25648  25672  25696  25720  25744 

1  5768  5792  5816  5840  5864  5888  5912  5935  5959  5983 

2  6007  6031  6055  6079  6102  6126  6150  6174  6198  6221 

3  6245  6269  6293  6316  6340  6364  6387  6411  6435  6458 

4  6482  6505  6529  6553  6576  6600  6623  6647  6670  6694 

5  6717  6741  6764  6788  6811  6834  6858  6881  6905  6928 

6  6951  6975  6998  7021  7045  7068  7091  7114  7138  7161 

7  7184  7207  7231  7254  7277  7300  7323  7346  7370  7393 

8  7416  7439  7462  7485  7508  7531  7554  7577  7600  7623 

9  7646  7669  7692  7715  7738  7761  7784  7807  7830  7852 

190  27875  27898  27921  27944  27967  27989  28012  28035  28058  28081 

1  8103  8126  8149  8171  8194  8217  8240  8262  8285  8307 

2  8330  8353  8375  8398  8421  8443  8466  8488  8511  8533 

3  8556  8578  8601  8623  8646  8668  8691  8713  8735  8758 

4  8780  8803  8825  8847  8870  8892  8914  8937  8959  8981 

5  9003  9026  9048  9070  9092  9115  9137  9159  9181  9203 

6  9226  9248  9270  9292  9314  933(5  9358  9380  9403  9425 

7  9447  9469  9491  9513  9535  9557  9579  9601  9623  9645 

8  9667  9688  9710  9732  9754  9776  9798  9820  9842  9863 

9  9885  9907  9929  9951  9973  9994  30016  30038  30060  30081 

200  30103  30125  30146  30168  30190  30211  30233  30255  30276  30298 

196 


TABLE   IX.  —  LOGARITHMS    OF   NUMBERS. 


3456789 


200 

1 
2 
3 
4 
5 


9 

210 

1 
2 
3 
4 
5 
6 
7 
8 
9 


3 
4 
5 
6 

7 
8 
9 

230 

1 
2 
3 
4 
5 
6 
7 
8 
9 


30103  30125  30146  30168  30190  30211  30233  30255  30276  30298 

0320  0341  0363  0384  0406  0428  0449  0471  0492  0514 

0535  0557  0578  0600  0621  0643  0664  0685  0707  0728 

0750  0771  0792  0814  0835  0856  0878  0899  0920  0942 

0963  0984  1006  1027  1048  1069  1091  1112  1133  1154 

1175  1197  1218  1239  1260  1281  1302  1323  1345  1366 

1387  1408  1429  1450  1471  1492  1513  1534  1555  1576 

1597  1618  1639  1660  1681  1702  1723  1744  1765  1785 

1806  1827  1848  1869  1890  1911  1931  1952  1973  1994 

2015  2035  2056  2077  2098  2118  2139  2160  2181  2201 

32222  32243  32263  32284  32305  32325  32346  32366  32387  32408 

2428  2449  2469  2490  2510  2531  2552  2572  2593  2613 

2634  2654  2675  2695  2715  2736  2756  2777  2797  2818 

2838  2858  2879  2899  2919  2940  2960  2980  3001  3021 

3041  3062  3082  3102  3122  3143  3163  3183  3203  3224 

3244  3264  3284  3304  3325  3345  3365  3385  3405  3425 

3445  3465  3486  3506  3526  3546  3566  3586  3606  3626 

3646  3666  3686  3706  3726  3746  3766  3786  3806  3826 

3846  3866  3885  3905  3925  3945  3965  3985  4005  4025 

4044  4064  4084  4104  4124  4143  4163  4183  4203  4223 

34242  34262  34282  34301  34321  34341  34361  34380  34400  34420 

4439  4459  4479  4498  4518  4537  4557  4577  4596  4616 

4635  4655  4674  4694  4713  4733  4753  4772  4792  4811 

4830  4850  4869  4889  4908  4928  4947  4967  4986  5005 

5025  5044  5064  5083  5102  5122  5141  5160  5180  5199 

5218  5238  5257  5276  5295  5315  5334  5353  5372  5392 

5411  5430  5449  5468  5488  5507  5526  5545  5564  5583 

5603  5622  5641  5660  5679  5698  5717  5736  5755  5774 

5793  5813  5832  5851  5870  5889  5908  5927  5946  5965 

5984  6003  6021  6040  6059  6078  6097  6116  6135  6154 

36173  36192  36211  36229  36248  36267  36286  36305  36324  36342 

6361  6380  6399  6418  6436  6455  6474  6493  6511  6530 

6549  6568  6586  6605  6624  6642  6661  6680  6698  6717 

6736  6754  6773  6791  6810  6829  6847  6866  6884  6903 

6922  6940  6959  6977  6996  7014  7033  7051  7070  7088 

7107  7125  7144  7162  7181  7199  7218  7236  7254  7273 

7291  7310  7328  7346  7365  7383  7401  7420  7438  7457 

7475  7493  7511  7530  7548  7566  7585  7603  7621  7639 

7658  7676  7694  7712  7731  7749  7767  7785  7803  7822 

7840  7858  7876  7894  7912  7931  7949  7967  7985  8003 

38021  38039  38057  38075  38093  38112  38130  38148  38166  38184 

8202  8220  8238  8256  8274  8292  8310  8328  8346  8364 

8382  8399  8417  8435  8453  8471  8489  8507  8525  8543 

8561  8578  8596  8614  8632  8650  8668  8686  8703  8721 

8739  8757  8775  8792  8810  8828  8846  8863  8881  8899 

8917  8934  8952  8970  8987  9005  9023  9041  9058  9076 

9094  9111  9129  9146  9164  9182  9199  9217  9235  9252 

9270  9287  9305  9322  9340  9358  9375  9393  9410  9428 

9445  9463  9480  9498  9515  9533  9550  9568  9585  9602 

9620  9637  9655  9672  9690  9707  9724  9742  9759  9777 


250  39794  39811  39829  39846  39863  39881  39898  39915  39933  39950 

197 


TABLE  IX.  —  LOGARITHMS   OF  NUMBERS. 


3 

4 
5 
6 
7 
8 
9 

260 
1 
2 
3 

4 
5 
6 
7 
8 
9 

270 
1 

2 
3 

4 
5 


280 
1 
2 
3 
4 
5 
6 
7 
8 
9 


678 


39794  39811  39829  39846  39803  39881  39898  39915  39933  39950 

9967  9985  40002  40019  40037  40054  40071  40088  40106  41)123 

4014040157  0175  0192  0209  0226  0243  0261  0278  0295 

0312  0329  0346  0364  0381  0398  0415  0432  0449  0466 

0483  0500  0518  0535  0552  0569  0586  0603  0620  0637 

0654  0671  0688  0705  0722  0739  0756  0773  0790  0807 

0824  0841  0858  0875  0892  0909  0926  0943  0960  0976 

0993  1010  1027  1044  1061  1078  1095  1111  1128  1145 

1162  1179  1196  1212  1229  1246  1263  1280  1296  1313 

1330  1347  1363  1380  1397  1414  1430  1447  1464  1481 

41497  41514  41531  41547  41564  41581  41597  41614  41631  41647 

1664  1681  1697  1714  1731  1747  1764  1780  1797  1814 

1830  1847  1863  1880  1896  1913  1929  1946  1963  1979 

1996  2012  2029  2045  2062  2078  2095  2111  2127  2144 

2160  2177  2193  2210  2226  2243  2259  2275  2292  2308 

2325  2341  2357  2374  2390  2406  2423  2439  2455  2472 

2488  2504  2521  2537  2553  2570  2586  2602  2619  2635 

2651  2667  2684  2700  2716  2732  2749  2765  2781  2797 

2813  2830  2846  2862  2878  2894  2911  2927  2943  2959 

2975  2991  3008  3024  3040  3056  3072  3088  3104  3120 

43136  43152  43169  43185  43201  43217  43233  43249  43265  43281 

3297  3313  3329  3345  3361  3377  3393  3409  3425  3441 

3457  3473  3489  3505  3521  3537  3553  3569  3584  3600 

3616  3632  3648  3664  3680  3696  3712  3727  3743  3759 

3775  3791  3807  3823  3838  3854  3870  3886  3902  3917 

3933  3949  3965  3981  3996  4012  4028  4044  4059  4075 

4091  4107  4122  4138  4154  4170  4185  4201  4217  4232 

4248  4264  4279  4295  4311  4326  4342  4358  4373  4389 

4404  4420  4436  4451  4467  4483  4498  4514  4529  4545 

4560  4576  4592  4607  4623  4638  4654  4669  4685  4700 

44716  44731  44747  44762  44778  44793  44809  44824  44840  44855 

4871  4886  4902  4917  4932  4948  4963  4979  4994  5010 

5025  5040  5056  5071  5086  5102  5117  5133  5148  5163 

5179  5194  5209  5225  5240  5255  5271  5286  5301  5317 

5332  5347  5362  5378  5393  5408  5423  5439  5454  5469 

5484  5500  5515  5530  5545  5561  5576  5591  5606  5621 

5637  5652  5667  5682  5697  5712  5728  5743  5758  5773 

5788  5803  5818  5834  5849  5864  5879  5894  5909  5924 

5939  5954  5969  5984  6000  6015  6030  6045  6060  6075 

6090  6105  6120  6135  6150  6165  6180  6195  6210  6225 

46240  46255  46270  46285  46300  46315  46330  46345  46359  46374 

6389  6404  6419  6434  6449  6464  6479  6494  6509  6523 

6538  6553  6568  6583  6598  6613  6627  6642  6657  6672 

6687  6702  6716  6731  6746  6761  6776  6790  6805  6820 

6835  6850  6864  6879  6894  6909  6923  6938  6953  6967 

6982  6997  7012  7026  7041  7056  7070"  7085  7100  7114 

7129  7144  7159  7173  7188  7202  7217  7232  7246  7261 

7276  7290  7305  7319  7334  7349  7363  7378  7392  7407 

7422  7436  7451  7465  7480  7494  7509  7524  7538  7553 

7567  7582  7596  7611  7625  7640  7654  7669  7683  7698 

47712  47727  47741  47756  47770  47784  47799  47813  47828  47842 


TABLE    IX.  —  LOGARITHMS    OF   NUMBERS. 


N 


0123456789 


4 
5 
6 
7 
8 
9 

310 
1 

2 
3 

4 
5 
6 
7 
8 
9 

320 

1 

2 
3 

4 
5 
6 

7 


9 


340 

1 
2 
3 

4 
5 
6 
7 
8 
9 


47712  47727  47741  47756  47770  47784  47799  47813  47828  47842 

7857  7871  7885  7900  7914  7929  7943  7958  7972  7986 

8001  8015  8029  8044  8058  8073  8087  8101  8116  8130 

8144  8159  8173  8187  8202  8216  8230  8244  8259  8273 

8287  8302  8316  8330  8344  8359  8373  8387  8401  8416 

8430  8444  8458  8473  8487  8501  8515  8530  8544  8558 

8572  8586  8601  8615  8629  8643  8657  8671  8686  8700 

8714  8728  8742  8756  8770  8785  8799  8813  8827  8841 

8855  8869  8883  8897  8911  8926  8940  8954  8968  8982 

8996  9010  9024  9038  9052  9066  9080  9094  9108  9122 

49136  49150  49164  49178  49192  49206  49220  49234  49248  49262 

9276  9290  9304  9318  9332  9346  9360  9374  9388  9402 

9415  9429  9443  9457  9471  9485  9499  9513  9527  9541 

9554  9568  9582  9596  9610  9624  9638  9651  9665  9679 

9693  970,7  9721  9734  9748  9762  9776  9790  9803  9817 

9831  9845  9859  9872  9886  9900  9914  9927  9941  9955 

9969  9982  9996  50010  50024  50037  50051  50065  50079  50092 

50106  50120  50133  0147  0161  0174  0188  0202  0215  0229 

0243  0256  0270  0284  0297  0311  0325  0338  0352  0365 

0379  0393  0406  0420  0433  0447  0461  0474  0488  0501 

50515  50529  50542  50556  50569  50583  50596  50610  50623  50637 

0651  0664  0678  0691  0705  0718  0732  0745  0759  0772 

0786  0799  0813  0826  0840  0853  0866  0880  0893  0907 

0920  0934  0947  0961  0974  0987  1001  1014  1028  1041 

1055  1068  1081  1095  1108  1121  1135  1148  1162  1175 

1188  1202  1215  1228  1242  1255  1268  1282  1295  1308 

1322  1335  1348  1362  1375  1388  1402  1415  1428  1441 

1455  1468  1481  1495  1508  1521  1534  1548  1561  1574 

1587  1601  1614  1627  1640  1654  1667  1680  1693  1706 

1720  1733  1746  1759  1772  1786  1799  1812  1825  1838 

51851  51865  51878  51891  51904  51917  51930  51943  51957  51970 

1983  1996  2009  2022  2035  2048  2061  2075  2088  2101 

2114  2127  2140  2153  2166  2179  2192  2205  2218  2231 

2244  2257  2270  2284  2297  2310  2323  2336  2349  2362 

2375  2388  2401  2414  2427  2440  2453  2466  2479  2492 

2504  2517  2530  2543  2556  2569  2582  2595  2608  2621 

2634  2647  2660  2673  2686  2699  2711  2724  2737  2750 

2763  2776  2789  2802  2815  2827  2840  2853  2866  2879 

2892  2905  2917  2930  2943  2956  2969  2982  2994  3007 

3020  3033  3046  3058  3071  3084  3097  3110  3122  3135 

53148  53161  53173  53186  53199  53212  53224  53237  53250  53263 

3275  3288  3301  3314  3326  3339  3352  3364  3377  3390 

3403  3415  3428  3441  3453  3466  3479  3491  3504  3517 

3529  3542  3555  3567  3580  3593  3605  3618  3631  3643 

3656  3668  3681  3694  3706  3719  3732  3744  3757  3769 

3782  3794  3807  3820  3832  3845  3857  3870  3882  3895 

3908  3920  3933  3945  3958  3970  3983  3995  4008  4020 

4033  4045  4058  4070  4083  4095  4108  4120  4133  4145 

4158  4170  4183  4195  4208  4220  4233  4245  4258  4270 

4283  4295  4307  4320  4:532  4345  4357  4370  4382  4394 

54407  54419  54432  54444  54456  54469  54481  54494  54506  54518 


TABLE   IX.  —  LOGARITHMS   OF   NUMBERS. 


56789 


54407  54419  54432  54444  54456  54469  54481  54494  54506  54518 

4531  4543  4555  4568  4580  4593  4605  4617  4630  4642 

4654  4667  4679  4691  4704  4716  4728  4741  4753  4765 

4777  4790  4802  4814  4827  4839  4851  4864  4876  4888 

4900  4913  4925  4937  4949  4962  4974  4986  4998  5011 

5023  5035  5047  5060  5072  5084  5096  5108  5121  5133 

5145  5157  5169  5182  5194  5206  5218  5230  5242  5255 

5267  5279  5291  5303  5315  5328  5340  5352  5364  5376 

5388  5400  5413  5425  5437  5449  5461  5473  5485  5497 

5509  5522  5534  5546  5558  5570  5582  5594  5606  5618 

55630  55642  55654  55666  55678  55691  55703  55715  55727  55739 

5751  5763  5775  5787  5799  5811  5823  5835  5847  5859 

5871  5883  5895  5907  5919  5931  5943  5955  5967  5979 

5991  6003  6015  6027  6038  6050  6062  6074  6086  6098 

6110  6122  6134  6146  6158  6170  6182  6194  6205  6217 

6229  6241  6253  6265  6277  6289  6301  6312  6324  6336 

6348  6360  6372  6384  6396  6407  6419  6431  6443  6455 

6467  6478  6490  6502  6514  6526  6538  6549  6561  6573 

6585  6597  6608  6620  6632  6644  6656  6667  6679  6691 

6703  6714  6726  6738  6750  6761  6773  6785  6797  6808 

56820  56832  56844  56855  56867  56879  56891  56902  56914  56926 

6937  6949  6961  6972  6984  6996  7008  7019  7031  7043 

7054  7066  7078  7089  7101  7113  7124  7136  7148  7159 

7171  7183  7194  7206  7217  7229  7241  7252  7264  7276 

7237  7299  7310  7322  7334  7345  7357  7368  7380  7392 

7403  7415  7426  7438  7449  7461  7473  7484  7496  7507 

7519  7530  7542  7553  7565  7576  7588  7600  7611  7623 

7634  7646  7657  7669  7680  7692  7703  7715  7726  7738 

7749  7761  7772  7784  7795  7807  7818  7830  7841  7852 

7864  7875  7887  7898  7910  7921  7933  7944  7955  7967 

57978  57990  58001  58013  58024  58035  58047  58058  58070  58081 

8092  8104  8115  8127  8138  8149  8161  8172  8184  8195 

8206  8218  8229  8240  8252  8263  8274  8286  8297  8309 

8320  8331  8343  8354  8365  8377  8388  8399  8410  8422 

8433  8444  8456  8467  8478  8490  8501  8512  8524  8535 

8546  8557  8569  8580  8591  8602  8614  8625  8636  8647 

8659  8670  8681  8692  8704  8715  8726  8737  8749  8760 

8771  8782  8794  8805  8816  8827  8838  8850  8861  8872 

8883  8894  8906  8917  8928  8939  8950  8961  8973  8984 

8995  9006  9017  9028  9040  9051  9062  9073  9084  9095 

59106  59118  59129  59140  59151  59162  59173  59184  59195  59207 

9218  9229  9240  9251  9262  9273  9284  9295  9306  9318 

9329  9340  9351  9362  9373  9384  9395  9406  9417  9428 

9439  9450  9461  9472  9483  9494  9506  9517  9528  9539 

9550  9561  9572  9583  9594  9605  9616  9627  9638  9649 

9660  9671  9632  9693  9704  9715  9726  9737  9748  9759 

9770  9780  9791  9802  9813  9824  9835  9846  9857  9868 

9879  9890  9901  9912  9923  9934  9945  9956  9966  9977 

9988  9999  60010  60021  60032  60043  60054  60065  60076  60086 

9  |  60097  60108  0119  0130  0141  0152  0163  0173  0184  0195 

400  i  60206  60217  60228  60239  60249  60260  60271  60282  60293  60304 


TABLE   IX.  —  LOGARITHMS    OF   NUMBERS. 


N0123456789 


60206  60217  60228  60239  60249  60260  60271  60282  60293  60304 

0314  0325  0336  0347  0358  0369  0379  0390  0401  0412 

0423  0433  0444  0455  0466  0477  0487  0498  0509  0520 

0531  0541  0552  0563  0574  0584  0595  0606  0617  0627 

4  0638  0649  0660  0670  0681  0692  0703  0713  0724  0735 

5  0746  0756  0767  0778  0788  0799  0810  0821  0831  0842 
0853  0863  0874  0885  0895  0906  0917  0927  0938  0949 

7   0959  0970  0981  0991  1002  1013  1023  1034  1045  1055 

1066  1077  1087  1098  1109  1119  1130  1140  1151  1162 

9   1172  1183  1194  1204  1215  1225  1236  1247  1257  1268 

61278  61289  61300  61310  61321  61331  61342  61352  61363  61374 

1384  1395  1405  1416  1426  1437  1448  1458  1469  1479 

1490  1500  1511  1521  1532  1542  1553  1563  1574  1584 

3  1595  1606  1616  1627  1637  1648  1658  1669  1679  1690 

4  1700  1711  1721  1731  1742  1752  1763  1773  1784  1794 

5  1805  1815  1826  1836  1847  1857  1868  1878  1888  1899 

6  1909  1920  1930  1941  1951  1962  1972  1982  1993  2003 

7  2014  2024  2034  2045  2055  2066  2076  2086  2097  2107 

8  2118  2128  2138  2149  2159  2170  2180  2190  2201  2211 

9  2221  2232  2242  2252  2263  2273  2284  2294  2304  2315 

420  62325  62335  62316  62356  62366  62377  62387  62397  62408  62418 

1  2428  2439  2449  2459  2469  2480  2490  2500  2511  2521 

2  2531  2542  2552  2562  2572  2583  2593  2603  2613  2624 

3  2634  2644  2655  2665  2675  2685  2696  2706  2716  2726 

4  2737  2747  2757  2767  2778  2788  2798  2808  2818  2829 

5  2839  2849  2859  2870  2880  2890  2900  2910  2921  2931 

6  2941  2951  2961  2972  2982  2992  3002  3012  3022  3033 

7  3043  3053  3063  3073  3083  3094  3104  3114  3124  3134 

8  3144  3155  3165  3175  3185  3195  3205  3215  3225  3236 

9  3246  3256  3266  3276  3286  3296  3306  3317  3327  3337 

63347  63357  63367  63377  63387  63397  63407  63417  63428  63438 

3448  3458  3468  3478  3488  3498  3508  3518  3528  3538 

3548  3558  3568  3579  3589  3599  3609  3619  3629  3639 

3649  3659  3669  3679  3689  3699  3709  3719  3729  3739 

4  3749  3759  3769  3779  3789  3799  3809  3819  3829  3839 

5  3849  3859  3809  3879  3889  3899  3909  3919  3929  3939 

6  3949  3959  3969  3979  3988  3998  4008  4018  4028  4038 

7  4048  4058  4068  4078  4088  4098  4108  4118  4128  4137 

8  4147  4157  4167  4177  4187  4197  4207  4217  4227  4237 

9  4246  4256  4266  4276  4286  4296  4306  4316  4326  4335 

64345  64355  64365  64375  64385  64395  64404  64414  64424  64434 

4444  4454  4464  4473  4483  4493  4503  4513  4523  4532 

2  4542  4552  4562  4572  4582  4591  4601  4611  4621  4631 

3  4640  4650  4660  4670  4680  4689  4699  4709  4719  4729 

4  4738  4748  4758  4768  4777  4787  4797  4807  4816  4826 

5  4836  4846  4856  4865  4875  4885  4895  4904  4914  4924 

6  4933  4943  4953  4963  4972  4982  4992  5002  5011  5021 

7  5031  5040  5050  5060  5070  5079  5089  5099  5108  5118 

8  5128  5137  5147  5157  5167  5176  5186  5196  5205  5215 

9  5225  5234  5244  5254  5263  5273  5283  5292  5302  5312 

450  i  65321  65331  65341  65350  65360  65369  65379  65389.65398  65408 

201 


TABLE   IX.  —  LOGARITHMS    OF   NUMBERS. 


6789 


65321  65331  65341  65350  65360  65369  65379  65389  65398  65408 

5418  5427  5437  5447  5456  5466  5475  5485  5495  5504 

5514  5523  5533  5543  5552  5562  5571  5581  5591  5600 

5610  5619  5629  5639  5648  5658  5667  5677  5686  5696 

5706  5715  5725  5734  5744  5753  5763  5772  5782  5792 

5801  5811  5820  5830  5839  5849  5858  5868  5877  5887 

5896  5906  5916  5925  5935  5944  5954  5963  5973  5982 

5992  6001  6011  6020  6030  6039  6049  6058  6068  6077 

6087  6096  6106  6115  6124  6134  6143  6153  6162  6172 

6181  6191  6200  6210  6219  6229  6238  6247  6257  6266 

66276  66285  66295  66304  66314  66323  66332  66342  66351  66361 

6370  6380  6389  6398  6408  6417  6427  6436  6445  6455 

6464  6474  6483  6492  6502  6511  6521  6530  6539  6549 

6558  6567  6577  6586  6596  6605  6614  6624  6633  6642 

6652  6661  6671  6680  6689  6699  6708  6717  6727  6736 

6745  6755  6764  6773  6783  6792  6801  6811  6820  6829 

6839  6848  6857  6867  6876  6885  6894  6904  6913  6922 

6932  6941  6950  6960  6969  6978  6987  6997  7006  7015 

7025  7034  7043  7052  7062  7071  7080  7089  7099  7108 

7117  7127  7136  7145  7154  7164  7173  7182  7191  7201 

67210  67219  67228  67237  67247  67256  67265  67274  67284  67293 

7302  7311  7321  7330  7339  7348  7357  7367  7376  7385 

7394  7403  7413  7422  7431  7440  7449  7459  7468  7477 

7486  7495  7504  7514  7523  7532  7541  7550  7560  7569 

7578  7587  7596  7605  7614  7624  7633  7642  7651  7660 

7669  7679  7688  7697  7706  7715  7724  7733  7742  7752 

7761  7770  7779  7788  7797  7806  7815  7825  7834  7843 

7852  7861  7870  7879  7888  7897  7906  7916  7925  7934 

7943  7952  7961  7970  7979  7988  7997  8006  8015  8024 

8034  8043  8052  8061  8070  8079  8088  8097  8106  8115 

68124  68133  68142  68151  68160  68169  68178  68187  68196  68205 

8215  8224  8233  8242  8251  8260  8269  8278  8287  8296 

8305  8314  8323  8332  8341  8350  8359  8368  8377  8386 

8395  8404  8413  8422  8431  8440  8449  8458  8467  8476 

8485  8494  8502  8511  8520  8529  8538  8547  8556  8565 

8574  8583  8592  8601  8610  8619  8628  8637  8646  8655 

8664  8673  8681  8690  8699  8708  8717  8726  8735  8744 

8753  8762  8771  8780  8789  8797  8806  8815  8824  8833 

8842  8851  8860  8869  8878  8886  8895  8904  8913  8922 

8931  8940  8949  8958  8966  8975  8984  8993  9002  9011 

69020  69028  69037  69046  69055  69064  69073  69082  69090  69099 

9108  9117  9126  9135  9144  9152  9161  9170  9179  9188 

9197  9205  9214  9223  9232  9241  9249  9258  9267  9276 

9285  9294  9302  9311  9320  9329  9338  9346  9355  9364 

9373  9381  9390  9399  9408  9417  9425  9434  9443  9452 

9461  9469  9478  9487  9496  9504  9513  9522  9531  9539 

9548  9557  9566  9574  9583  9592  9601  9609  9618  9627 

9636  9644  9653  9662  9671  9679  9688  9697  9705  9714 

9723  9732  9740  9749  9758  9767  9775  9784  9793  9801 

9810  9819  9827  9836  9845  9854  9862  9871  9880  9888 

500  69897  69906  69914  69923  69932  69940  69949  69958  69966  69975 

202 


TABLE   IX.  —  LOGARITHMS    OF   NUMBERS. 


N0123456789 


500 

1 
2 
3 
4 
5 
6 
7 
8 
9 


520 

1 
2 
3 

4 
5 
6 

7 


9 


7 
8 
9 

540 

1 
2 
3 
4 
5 
6 
7 
8 
9 


69897  69906  69914  69923  699:52  69940  69949  69958  69966  69975 

9984  9992  70001  70010  70018  70027  70036  70044  70053  70062 

70070  70079  0088  0096  0105  0114  0122  0131  0140  0148 

0157  0165  0174  0183  0191  0200  0209  0217  0226  0234 

0243  0252  026;)  0269  0278  0286  0295  0303  0312  0321 

0329  0338  0346  0355  0364  0372  0381  0389  0398  0406 

0415  0424  0432  0441  0449  0458  0467  0475  0484  0492 

0501  0509  0518  0526  0535  0544  0552  0561  0569  0578 

0586  0595  0603  0612  0621  0629  0638  0646  0655  0663 

0672  0680  0689  0697  0706  0714  0723  0731  0740  0749 

70757  70766  70774  70783  70791  70800  70808  70817  70825  70834 

0842  0851  0859  0868  0876  0885  0893  0902  0910  0919 

0927  0935  0944  0952  0961  0969  0978  0986  0995  1003 

1012  1020  1029  1037  1046  1054  1063  1071  1079  1088 

1096  1105  1113  1122  1130  1139  1147  1155  1164  1172 

1181  1189  1198  1206  1214  1223  1231  1240  1248  1257 

1265  1273  1282  1290  1299  1307  1315  1324  1332  1341 

1349  1357  1366  1374  1383  1391  1399  1408  1416  1425 

1433  1441  1450  1458  1466  1475  1483  1492  1500  1508 

1517  1525  1533  1542  1550  1559  1567  1575  1584  1592 

71600  71609  71617  71625  71634  71642  71650  71659  71667  71675 

1684  1692  1700  1709  1717  1725  1734  1742  1750  1759 

1767  1775  1784  1792  1800  1809  1817  1825  1834  1842 

1850  1858  1867  1875  1883  1892  1900  1908  1917  1925 

1933  1941  1950  1958  1966  1975  1983  1991  1999  2008 

2016  2024  2032  2041  2049  2057  2066  2074  2082  2090 

2099  2107  2115  2123  2132  2140  2148  2156  2165  2173 

2181  2189  2198  2206  2214  2222  2230  2239  2247  2255 

2263  2272  2280  2288  2296  2304  2313  2321  2329  2337 

2346  2354  2362  2370  2378  2387  2395  2403  2411  2419 

72428  72436  72444  72452  72460  72469  72477  72485  72493  72501 

2509  2518  2526  2534  2542  2550  2558  2567  2575  2583 

2591  2599  2607  2616  2624  2632  2640  2648  2656  2665 

2673  2681  2689  2697  2705  2713  2722  2730  2738  2746 

2754  2762  2770  2779  2787  2795  2803  2811  2819  2827 

2835  2843  2852  2860  2868  2876  2884  2892  2900  2908 

2916  2925  2933  2941  2949  2957  2965  2973  2981  2989 

2997  3006  3014  3022  3030  3038  3046  3054  3062  3070 

3078  3086  3094  3102  3111  3119  3127  3135  3143  3151 

3159  3167  3175  3183  3191  3199  3207  3215  3^23  3231 

73239  73247  73255  73263  73272  73280  73288  73296  73304  73312 

3320  3328  3336  3344  3352  3360  3368  3376  3384  3392 

3400  3408  3416  3424  3432  3440  3448  3456  3464  3472 

3480  3488  3496  3504  3512  3520  3528  3536  3544  3552 

3560  3568  3576  3584  3592  3600  3608  3616  3624  3632 

3640  3648  3656  3664  3672  3679  3687  3695  3703  3711 

3719  3727  3735  3743  3751  3759  3767  3775  3783  3791 

3799  3807  3815  3823  3830  3838  3846  3854  3862  3870 

3878  3886  3894  3902  3910  3918  3926  3933  3941  3949 

3957  3965  3973  3981  3989  3997  4005  4013  4020  4028 

74036  74044  74052  74060  74068  74076  74084  74092  74099  74107 


TABLE   IX.  —  LOGARITHMS   OF   NUMBERS. 


NO123456789 


550 

1 
2 
3 
4 
5 
6 
7 


9 


4 
5 
6 
7 
8 
9 

570 

1 

2 
3 

4 
5 
6 

7 
8 
9 

580 

1 
2 
3 

4 
5 
6 
7 
8 
9 


74036  74044  74052  74060  74068  74076  74084  74092  74099  74107 

4115  4123  4131  4139  4147  4155  4162  4170  4178  4186 

4194  4202  4210  4218  4225  4233  4241  4249  4257  4265 

4273  4280  4288  4296  4304  4312  4320  4327  4335  4343 

4351  4359  4367  4374  4382  4390  4398  4406  4414  4421 

4429  4437  4445  4453  4461  4468  4476  4484  4492  4500 

4507  4515  4523  4531  4539  4547  4554  4562  4570  4578 

4586  4593  4601  4609  4617  4624  4632  4640  4648  4656 

4663  4671  4679  4687  4695  4702  4710  4718  4726  4733 

4741  4749  4757  4764  4772  4780  4788  4796  4803  4811 

74819  74827  74834  74842  74850  74858  74865  74873  74881  74889 

4896  4904  4912  4920  4927  4935  4943  4950  4958  4966 

4974  4981  4989  4997  5005  5012  5020  5028  5035  5043 

5051  5059  5066  5074  5082  5089  5097  5105  5113  5120 

5128  5136  5143  5151  5159  5166  5174  5182  5189  5197 

5205  5213  5220  5228  5236  5243  5251  5259  5266  5274 

5282  5289  5297  5305  5312  5320  5328  5335  5343  5351 

5358  5366  5374  5381  5389  5397  5404  5412  5420  5427 

5435  5442  5450  5458  5465  5473  5481  5488  5496  5504 

5511  5519  5526  5534  5542  5549  5557  5565  5572  5580 

75587  75595  75603  75610  75618  75626  75633  75641  75648  75656 

5664  5671  5679  5686  5694  5702  5709  5717  5724  5732 

5740  5747  5755  5762  5770  5778  5785  5793  5800  5808 

5815  5823  5831  5838  5846  5853  5861  5868  5876  5884 

5891  5899  5906  5914  5921  5929  5937  5944  5952  5959 

5967  5974  5982  5989  5997  6005  6012  6020  6027  6035 

6042  6050  6057  6065  6072  6080  6087  6095  6103  6110 

6118  6125  6133  6140  6148  6155  6163  6170  6178  6185 

6193  6200  6208  6215  6223  6230  6238  6245  6253  6260 

6268  6275  6283  6290  6298  6305  6313  6320  6328  6335 

76343  76350  76358  76365  76373  76380  76388  76395  76403  76410 

6418  6425  6433  6440  6448  6455  6462  6470  6477  6485 

6492  6500  6507  6515  6522  6530  6537  6545  6552  6559 

6567  6574  6582  6589  6597  6604  6612  6619  6626  6634 

6641  6649  6656  6664  6671  6678  6686  6693  6701  6708 

6716  6723  6730  6738  6745  6753  6760  6768  6775  6782 

6790  6797  6805  6812  6819  6827  6834  6842  6849  6856 

6864  6871  6879  6886  6893  6901  6908  6916  6923  6930 

6938  6945  6953  6960  6967  6975  6982  6989  6997  7004 

7012  7019  7026  7034  7041  7048  7056  7063  7070  7078 

77085  77093  77100  77107  77115  77122  77129  77137  77144  77151 

7159  7166  7173  7181  7188  7195  7203  7210  7217  7225 

7232  7240  7247  7254  7262  7269  7276  7283  7291  7298 

7305  7313  7320  7327  7335  7342  7349  7357  7364  7371 

7379  7386  7393  7401  7408  7415  7422  7430  7437  7444 

7452  7459  7466  7474  7481  7488  7495  7503  7510  7517 

7525  7532  7539  7546  7554  7561  7568  7576  7583  7590 

7597  7605  7612  7619  7627  7634  7641  7648  7656  7663 

7670  7677  7685  7692  7699  7706  7714  7721  7728  7735 

7743  7750  7757  7764  7772  7779  7786  7793  7801  7808 

77815  77822  77830  77837  77844  77851  77859  77866  77873  77880 
2(H 


TABLE    IX.  —  LOGARITHMS    OF   NUMBERS. 


6789 


600 

1 


77815  77822  77830  77837  77844  77851  77859  77866  77873  77880 
7887  7895  7902  7909  7916  7924  7931  7938  7945  7952 


2  7960  7967  7974  7981  7988  7996  8003  8010  8017  8025 

3  8032  8039  8046  8053  8061  8068  8075  8082  8089  8097 


8104  .8111  8118  8125  8132  8140  8147  8154  8161  8168 

8176  8183  8190  8197  8204  8211  8219  8226  8233  8240 

8247  8254  8262  8269  8276  8283  8290  8297  8305  8312 

8319  8326  8333  8340  8347  8355  8362  8369  8376  8383 

8390  8398  8405  8412  8419  8426  8433  8440  8447  8455 

8462  8469  8476  8483  8490  8497  8504  8512  8519  8526 

78533  78540  78547  78554  78561  78569  78576  78583  78590  78597 

8604  8611  8618  8625  8633  8640  8647  8654  8661  8668 

8675  8682  8689  8696  8704  8711  8718  8725  8732  8739 


1 
2 

3  \  8746  8753  8760  8767  8774  8781  8789  8796  8803  8810 


8817  8824  8831  8838  8845  8852  8859  8866  8873  8880 

8888  8895  8902  8909  8916  8923  8930  8937  8944  8951 

8958  8965  8972  8979  8986  8993  9000  9007  9014  9021 

9029  9036  9043  9050  9057  9064  9071  9078  9085  9092 

9099  9106  9113  9120  9127  9134  9141  9148  9155  9162 

9169  9176  9183  9190  9197  9204  9211  9218  9225  9232 

79239  79246  79253  79260  79267  79274  79281  79288  79295  79302 

9309  9316  9323  9330  9337  9341  9351  9358  9365  9372 

9379  9386  9393  9400  9407  9414  9421  9428  9435  9442 

9449  9456  9463  9470  9477  9484  9491  9498  9505  9511 

9518  9525  9532  9539  9546  9553  95(50  9567  9574  9581 

9588  9595  9602  9609  9616  9623  9630  9637  9644  9650 

9657  9664  9671  9678  9685  9692  9699  9706  9713  9720 

9727  9734  9741  9748  9754  9761  9768  9775  9782  9789 

9796  9803  9810  9817  9824  9831  9837  9844  9851  9858 

9865  9872  9879  9886  9893  9900  9906  9913  9920  9927 

79934  79941  79948  79955  79962  79969  79975  79982  79989  79996 

80003  80010  80017  80024  80030  80037  80044  80051  80058  80065 

0072  0079  0085  0092  0099  0106  0113  0120  0127  0134 

0140  0147  0154  0161  0168  0175  0182  0188  0195  0202 

0209  0216  0223  0229  0236  0243  0250  0257  0264  0271 

0277  0284  0291  0298  0305  0312  0318  0325  0332  0339 

0346  0353  0359  0366  0373  0380  0387  0393  0400  0407 

0414  0421  0428  0434  0441  0448  0455  0462  0468  0475 

0482  0489  0496  0502  0509  0516  0523  0530  0536  0543 

0550  0557  0564  0570  0577  0584  0591  0598  0604  0611 

80618  80625  80632  80638  80645  80652  80659  80665  80672  80679 

0686  06D3  0699  0706  0713  0720  0726  0733  0740  0747 

0754  0760  0767  0774  0781  0787  0794  0801  0808  0814 

0821  0828  0835  0841  0848  0855  0862  0868  0875  0882 

0895  0002  0909  0916  0922  0929  0936  0943  0949 


0956  0963  0969  0976  0983  0990  0996  1003  1010  1017 

1023  1030  1037  1043  1050  1057  1064  1070  1077  1084 

1090  1097  1104  1111  1117  1124  1131  1137  1144  1151 

1158  1164  1171  1178  1184  1191  1198  1204  1211  1218 

1224  1231  1238  1245  1251  1258  1265  1271  1278  1285 


050  81291  81298  81305  81311  81318  81325  81331  81338  81345  81351 

iius 


TABLE   IX.  —  LOGARITHMS    OF    NUMBERS. 


0123456789 


(>50 
1 
2 
3 
4 
5 
6 
7 


680 

1 
2 
3 

4 
5 
6 

7 


9 


81291  81298  81305  81311  81318  81325  81331  81338  81345  81351 

1358  1365  1371  1378  1385  1391  1398  1405  1411  1418 

1425  1431  1438  1445  1451  1458  1465  1471  1478  1485 

1491  1498  1505  1511  1518  1525  1531  1538  1544  1551 

1558  1564  1571  1578  1584  1591  1598  1604  1611  1617 

1624  1631  1637  1644  1651  1657  1664  1671  1677  1684 

1690  1697  1704  1710  1717  1723  1730  1737  1743  1750 

1757  1763  1770  1776  1783  1790  1796  1803  1809  1816 

1823  1829  1836  1842  1849  1856  1862  1869  1875  1882 

1889  1895  1902  1908  1915  1921  1928  1935  1941  1948 

81954  81961  81968  81974  81981  81987  81994  82000  82007  82014 

2020  2027  2033  2040  2046  2053  2060  2066  2073  2079 

2086  2092  2099  2105  2112  2119  2125  2132  2138  2145 

2151  2158  2164  2171  2178  2184  2191  2197  2204  2210 

2217  2223  2230  2236  2243  2249  2256  2263  2269  2276 

2282  2289  2295  2302  2308  2315  2321  2328  2334  2341 

2347  2354  2360  2367  2373  2380  2387  2393  2400  2406 

2413  2419  2426  2432  2439  2445  2452  2458  2465  2471 

2478  2484  2491  2497  2504  2510  2517  2523  2530  2536 

2543  2549  2556  2562  2569  2575  2582  2588  2595  2601 

82607  82614  82620  82627  82633  82640  82646  82653  82659  82666 

2672  2679  2685  2692  2698  2705  2711  2718  2724  2730 

2737  2743  2750  2756  2763  2769  2776  2782  2789  2795 

2802  2808  2814  2821  2827  2834  2840  2847  2853  2860 

2866  2872  2879  2885  2892  2898  2905  2911  2918  2924 

2930  2937  2943  2950  2956  2963  2969  2975  2982  2988 

2995  3001  3008  3014  3020  3027  3033  3040  3046  3052 

3059  3065  3072  3078  3085  3091  3097  3104  3110  3117 

3123  3129  3136  3142  3149  3155  3161  3168  3174  3181 

3187  3193  3200  3206  3213  3219  3225  3232  3238  3245 

83251  83257  83264  83270  83276  83283  83289  83296  83302  83308 

3315  3321  3327  3334  3340  3347  3353  3359  3366  3372 

3378  3385  3391  3398  3404  3410  3417  3423  3429  3436 

3442  3448  3455  3461  3467  3474  3480  3487  3493  3499 

3506  3512  3518  3525  3531  3537  3544  3550  3556  3503 

3569  3575  3582  3588  3594  3601  3607  3613  3620  3626 

3632  3639  3645  3651  3658  3664  3670  3677  3683  3689 

3702  3708  3715  3721  3727  3734  3740  3746  3753 

3759  3765  3771  3778  3784  3790  3797  3803  3809  3816 

3822  3828  3835  3841  3847  3853  3860  3866  3872  3879 

83885  83891  83897  83904  83910  83916  83923  83929  83935  83942 

3948  3954  3960  3967  3973  3979  3985  3992  3998  4004 

4011  4017  4023  4029  4036  4042  4048  4055  4061  4067 

4073  4080  4086  4092  4098  4105  4111  4117  4123  4130 

4136  4142  4148  4155  4161  4167  4173  4180  4186  4192 

4198  4205  4211  4217  4223  4230  4236  4242  4248  4255 

4261  4267  4273  4280  4286  4292  4298  4305  4311  4317 

4323  4330  4336  4342  4348  4354  4361  4367  4373  4379 

4386  4392  4398  4404  4410  4417  4423  4429  4435  4442 

4448  4454  4460  4466  4473  4479  4485  4491  4497  4504 


TABLE   IX.  —  LOGARITHMS   OF   NUMBERS. 


NO123456789 


00 

1 
2 
3 
4 
5 
6 
7 
8 
9 

710 

1 
2 
3 
4 
5 
6 
7 
8 
9 

720 

1 
2 
3 
4 
5 
6 
7 
8 
9 

730 

1 

2 
3 

4 
5 
6 

7 
8 
9 

740 

1 
2 

3 
4 
5 
6 
7 
8 
9 


84510  84516  84522  84528  84535  84541  84547  84553  84559  84566 

4572  4578  4584  4590  4597  4603  4609  4615  4621  4628 

4634  4640  4646  4652  4658  4665  4671  4677  4683  4689 

4696  4702  4708  4714  4720  4726  4733  4739  4745  4751 

4757  4763  4770  4776  4782  4788  4794  4800  4807  4813 

4819  4825  4831  4837  4844  4850  4856  4862  4868  4874 

4880  4887  4893  4899  4905  4911  4917  4924  4930  4936 

4942  4948  4954  4960  4967  4973  4979  4985  4991  4997 

5003  5009  5016  5022  5028  5034  5040  5046  5052  5058 

5065  5071  5077  5083  5089  5095  5101  5107  5114  5120 

85126  85132  85138  85144  85150  85156  85163  85169  85175  85181 

5187  5193  5199  5205  5211  5217  5224  5230  5236  5242 

5248  5254  5260  5266  5272  5278  5285  5291  5297  5303 

5309  5315  5321  5327  5333  5339  5345  5352  5358  5364 

5370  5376  5382  5388  5394  5400  5406  5412  5418  5425 

5431  5437  5443  5449  5455  5461  5467  5473  5479  5485 

5491  5497  5503  5509  5516  5522  5528  5534  5540  5546 

5552  5558  5564  5570  5576  5582  5588  5594  5600  5606 

5612  5618  5625  5631  5637  5643  5649  5655  5661  5667 

5673  5679  5685  5691  5697  5703  5709  5715  5721  5727 

85733  85739  85745  85751  85757  85763  85769  85775  85781  85788 

5794  5800  5806  5812  5818  5824  5830  5836  5842  5848 

5854  5860  5866  5872  5878  5884  5890  5896  5902  5908 

5914  5920  5926  5932  5938  5944  5950  5956  5962  5968 

5974  5980  5986  5992  5998  6004  6010  6016  6022  6028 

6034  6040  6046  6052  6058  6064  6070  6076  6082  6088 

6094  6100  6106  6112  6118  6124  6130  6136  6141  6147 

6153  6159  6165  6171  6177  6183  6189  6195  6201  6207 

6213  6219  6225  6231  6237  6243  6249  6255  6261  6267 

6273  6279  6285  6291  6297  6303  6308  6314  6320  6326 

86332  86338  86344  86350  86356  86362  86368  86374  86380  86386 

6392  6398  6404  6410  6415  6421  6427  6433  6439  6445 

6451  6457  6463  6469  6475  6481  6487  6493  6499  6504 

6510  6516  6522  6528  6534  6540  6546  6552  6558  6564 

6570  6576  6581  6587  6593  6599  6605  6611  6617  6623 

6629  6635  6641  6646  6652  6658  6664  6670  6676  6682 

6688  6694  6700  6705  6711  6717  6723  6729  6735  6741 

6747  6753  6759  6764  6770  6776  6782  6788  6794  6800 

6806  6812  6817  6823  6829  6835  6841  6847  6853  6859 

6864  6870  6876  6882  6888  6894  6900  6906  6911  6917 

86923  86929  86935  86941  86947  86953  86958  86964  86970  86976 

6982  6988  6994  6999  7005  7011  7017  7023  7029  7035 

7040  7046  7052  7058  7064  7070  7075  7081  7087  7093 

7099  7105  7111  7116  7122  7128  7134  7140  7146  7151 

7157  7163  7169  7175  7181  7186  7192  7198  7204  7210 

7216  7221  7227  7233  7239  7245  7251  7256  7262  7268 

7274  7280  7286  7291  7297  7303  7309  7315  7320  7326 

7332  7338  7344  7349  7355  7361  7367  7373  7379  7384 

7390  7396  7402  7408  7413  7419  7425  7431  7437  7442 

7448  7454  7460  7466  7471  7477  7483  7489  7495  7500 

87506  87512  87518  87523  87529  87535  87541  87547  87552  87558 
207 


TABLE   IX.  —  LOGARITHMS   OF   NUMBERS. 


STO12345678 


790 

1 
2 
3 
4 
5 
6 
7 
8 
9 


87506  87512  87518  87523  87529  87535  87541  87547  87552  87558 

7564  7570  7576  7581  7587  7593  7599  7604  7610  7616 

7622  7628  7633  7639  7645  7651  7656  7662  7668  7674 

7679  7685  7691  7697  7703  7708  7714  7720  7726  7731 

7737  7743  7749  7754  7760  7766  7772  7777  7783  7789 

7795  7800  7806  7812  7818  7823  7829  7835  7841  7846 

7852  7858  7864  7869  7875  7881  7887  7892  7898  7904 

7910  7915  7921  7927  7933  7938  7944  7950  7955  7961 

7967  7973  7978  7984  7990  7996  8001  8007  8013  8018 

8024  8030  8036  8041  8047  8053  8058  8064  8070  8076 

88081  88087  88093  88098  88104  88110  88116  88121  88127  88133 

8138  8144  8150  8156  8161  8167  8173  8178  8184  8190 

8195  8201  8207  8213  8218  8224  8230  8235  8241  8247 

8252  8258  8264  8270  8275  8281  8287  8292  8298  8304 

8309  8315  8321  8326  8332  8338  8343  8349  8355  8360 

8366  8372  8377  8383  8389  8395  8400  8406  8412  8417 

8423  8429  8434  8440  8446  8451  8457  8463  8468  8474 

8480  8485  8491  8497  8502  8508  8513  8519  8525  8530 

8536  8542  8547  8553  8559  8564  8570  8576  8581  8587 

8593  8598  8604  8610  8615  8621  8627  8632  8638  8643 

88649  88655  88660  88666  88672  88677  88683  88689  88694  88700 

8705  8711  8717  8722  8728  8734  8739  8745  8750  8756 

8762  8767  8773  8779  8784  8790  8795  8801  8807  8812 

8818  8824  8829  8835  8840  8846  8852  8857  8863  8868 

8874  8880  8885  8891  8897  8902  8908  8913  8919  8925 

8930  8936  8941  8947  8953  8958  8964  8969  8975  8981 

8988  8992  8997  9003  9009  9014  9020  9025  9031  9037 

9042  9048  9053  9059  9064  9070  9076  9081  9087  9092 

9098  9104  9109  9115  9120  9126  9131  9137  9143  9148 

9154  9159  9165  9170  9176  9182  9187  9193  9198  9204 

89209  89215  89221  89226  89232  89237  89243  89248  89254  89260 

9265  9271  9276  9282  9287  9293  9298  9304  9310  9315 

9321  9326  9332  9337  9343  9348  9354  9360  9365  9371 

9376  9382  9387  9393  9398  9404  9409  9415  9421  9426 

9432  9437  9443  9448  9454  9459  9465  9470  9476  9481 

9487  9492  9498  9504  9509  9515  9520  9526  9531  9537 

9542  9548  9553  9559  9564  9570  9575  9581  9586  9f,<)2 

9597  9603  9609  9614  9620  9625  9631  9636  9642  9047 

9653  9658  9664  9669  9675  9680  9686  9691  9697  9702 

9708  9713  9719  9724  9730  9735  9741  9746  9752  9757 

89763  89768  89774  89779  89785  89790  89796  89801  89807  89812 

9818  9823  9829  9834  9840  9845  9851  9856  9862  9867 

9873  9878  9883  9889  9894  9900  9905  9911  9916  9922 

9927  9933  9938  9944  9949  9955  9960  9966  9971  9977 

9982  9988  9993  9998  90004  90009  90015  90020  90026  90031 

90037  90042  90048  90053  0059  0064  0069  0075  0080  0086 

0091  0097  0102  0108  0113  0119  0124  0129  0135  0140 

0146  0151  0157  0162  0168  0173  0179  0184  0189  0195 

0200  0206  0211  0217  0222  0227  0233  0238  0244  0249 

0255  0260  0266  0271  0276  0282  0287  0293  0298  0304 

90309  90314  90320  90325  90331  90336  90342  90347  90352  90358 

208 


TABLE   IX.  —  LOGARITHMS    OF   NUMBERS. 


56789 


810 

1 
2 
3 

4 
5 
6 

7 
8 
9 

820 
1 
2 
3 
4 
5 


90309  90314  90320  90325  90331  90336  90342  90347  90352  90358 
0363  0369  0374  0380  0385  0390  0396  0401  0407  0412 
0417  0423  0428  0434  0439  0445  0450  0455  0461  0466 
0472  0477  0482  0488  0493  0499  0504  0509  0515  0520 
0526  0531  0536  0542  0547  0553  0558  0563  0569  0574 
0580  0585  0590  0596  0601  0607  0612  0617  0623  0628 


6   0634  0639  0644  0650  0655  0660  0666  0671  0677  0682 


0687  0693  0698  0703  0709  0714  0720  0725  0730  0736 

0741  0747  0752  0757  0763  0768  0773  0779  0784  0789 

0795  0800  0806  0811  0816  0822  0827  0832  0838  0843 

90849  90854  90859  90865  90870  90875  90881  90886  90891  90897 

0902  0907  0913  0918  0924  0929  0934  0940  0945  0950 

0956  0961  0966  0972  0977  0982  0988  0993  0998  1004 

1009  1014  1020  1025  1030  1036  1041  1046  1052  1057 

1062  1068  1073  1078  1084  1089  1094  1100  1105  1110 

1116  1121  1126  1132  1137  1142  1148  1153  1158  1164 

1169  1174  1180  1185  1190  1196  1201  1206  1212  1217 

1222  1228  1233  1238  1243  1249  1254  1259  1265  1270 

1275  1281  1286  1291  1297  1302  1307  1312  1318  1323 

1328  1334  1339  1344  1350  1355  1360  1365  1371  1376 

91381  91387  91392  91397  91403  91408  91413  91418  91424  91429 

1434  1440  1445  1450  1455  1461  1466  1471  1477  1482 

1487  1492  1498  1503  1508  1514  1519  1524  1529  1535 

1540  1545  1551  1556  1561  1566  1572  1577  1582  1587 

1593  1598  1603  1609  1614  1619  1624  1630  1635  1640 

1645  1651  1656  1661  1666  1672  1677  1682  1687  1693 

1698  1703  1709  1714  1719  1724  1730  1735  1740  1745 

1751  1756  1761  1766  1772  1777  1782  1787  1793  1798 

1803  1808  1814  1819  1824  1829  1834  1840  1845  1850 

1855  1861  1866  1871  1876  1882  1887  1892  1897  1903 

91908  91913  91918  91924  91929  91934  91939  91944  91950  91955 

I960  1965  1971  1976  1981  1986  1991  1997  2002  2007 

2012  2018  2023  2028  2033  2038  2044  2049  2054  2059 

2065  2070  2075  2080  2085  2091  2096  2101  2106  2111 

2117  2122  2127  2132  2137  2143  2148  2153  2158  2163 

2169  2174  2179  2184  2189  2195  2200  2205  2210  2215 

2221  2226  2231  2236  2241  2247  2252  2257  2262  2267 

2273  2278  2283  2288  2293  2298  2304  2309  2314  2319 

2324  2330  2335  2340  2345  2350  2355  2361  2366  2371 

2376  2381  2387  2392  2397  2402  2407  2412  2418  2423 

92428  92433  92438  92443  92449  92454  92459  92464  92469  92474 

2480  2485  2490  2495  2500  2505  2511  2516  2521  2526 

2531  2536  2542  2547  2552  2557  2562  2567  2572  2578 

2583  2588  2593  2598  2603  2609  2614  2619  2624  2629 

2634  2639  2645  2650  2655  2660  2665  2670  2675  2681 

2686  2691  2696  2701  2706  2711  2716  2722  2727  2732 

2737  2742  2747  2752  2758  2763  2768  2773  2778  2783 

2788  2793  2799  2804  2809  2814  2819  2824  2829  2834 

2840  2845  2850  2855  2860  2865  2870  2875  2881  2886 


2891  2896  2901  2906  2911  2916  2921  2927  2932  2937 
850  92942  92947  92952  92957  92962  92967  92973  92978  92983  92988 

"209 


TABLE   IX.  —  LOGARITHMS    OF   NUMBERS. 


NO123456789 


850 

1 

2 
3 

4 
5 
6 

7 


860 

1 
2 
3 
4 
5 
6 
7 


9 


92942  92947  92952  92957  92962  92967  92973  92978  92983  92988 

2993  2998  3003  3008  3013  3018  3024  3029  3034  3039 

3044  3049  3054  3059  3064  3069  3075  3080  3085  3090 

3095  3100  3105  3110  3115  3120  3125  3131  3136  3141 

3146  3151  3156  3161  3166  3171  3176  3181  3186  3192 

3197  3202  3207  3212  3217  3222  3227  3232  3237  3242 

3247  3252  3258  3263  3268  3273  3278  3283  3288  3293 

3298  3303  3308  3313  3318  3323  3328  3334  3339  3344 

3349  3354  3359  3364  3369  3374  3379  3384  3389  3394 

3399  3404  3409  3414  3420  3425  3430  3435  3440  3445 

93450  93455  93460  93465  93470  93475  93480  93485  93490  93495 

3500  3505  3510  3515  3520  3526  3531  3536  3541  3546 

3551  3556  3561  3566  3571  3576  3581  3586  3591  3596 

3601  3606  3611  3616  3621  3626  3631  3636  3641  3646 

3651  3656  3661  3666  3671  3676  3682  3687  3692  3697 

3702  3707  3712  3717  3722  3727  3732  3737  3742  3747 

3752  3757  3762  3767  3772  3777  3782  3787  3792  3797 

3802  3807  3812  3817  3822  3827  3832  3837  3842  3847 

3852  3857  3862  3867  3872  3877  3882  3887  3892  3897 

3902  3907  3912  3917  3922  3927  3932  3937  3942  3947 

93952  93957  93962  93967  93972  93977  93982  93987  93992  93997 

4002  4007  4012  4017  4022  4027  4032  4037  4042  4047 

4052  4057  4062  4067  4072  4077  4082  4086  4091  4096 

4101  4106  4111  4116  4121  4126  4131  4136  4141  4146 

4151  4156  4161  4166  4171  4176  4181  4186  4191  4196 

4201  4206  4211  4216  4221  4226  4231  4236  4240  4245 

4250  4255  4260  4265  4270  4275  4280  4285  4290  4295 

4300  4305  4310  4315  4320  4325  4330  4335  4340  4345 

4349  4354  4359  4364  4369  4374  4379  4384  4389  4394 

4399  4404  4409  4414  4419  4424  4429  4433  4438  4443 

94448  94453  94458  94463  94468  94473  94478  94483  94488  94493 

4498  4503  4507  4512  4517  4522  4527  4532  4537  4542 

4547  4552  4557  4562  4567  4571  4576  4581  4586  4591 

4596  4601  4606  4611  4616  4621  4626  4630  4635  4640 

4645  4650  4655  4660  4665  4670  4675  4680  4685  4689 

4694  4699  4704  4709  4714  4719  4724  4729  4734  4738 

4743  4748  4753  4758  4763  4768  4773  4778  4783  4787 

4792  4797  4802  4807  4812  4817  4822  4827  4832  4836 

4841  4846  4851  4856  4861  4866  4871  4876  4880  4885 

4890  4895  4900  4905  4910  4915  4919  4924  4929  4934 

94939  94944  94949  94954  94959  94963  94968  94973  94978  94983 

4988  4993  4998  5002  5007  5012  5017  5022  5027  5032 

5036  5041  5046  5051  5056  5061  5066  5071  5075  5080 

5085  5090  5095  5100  5105  5109  5114  5119  5124  5129 

5134  5139  5143  5148  5153  5158  5163  5168  5173  5177 

5182  5187  5192  5197  5202  5207  5211  5216  5221  5226 

5231  5236  5240  5245  5250  5255  5260  5265  5270  5274 

5279  5284  5289  5294  5299  5303  5308  5313  5318  5323 

5328  5332  5337  5342  5347  5352  5357  5361  5366  5371 

5376  5381  5386  5390  5395  5400  5405  5410  5415  5419 

95424  95429  95434  95439  95444  95448  95453  95458  95463  95468 


TABLE   IX.  —  LOGARITHMS    OF   NUMBERS. 


0123456789 


95424  95429  95434  95439  95444  95448  95453  95458  95463  95468 

5472  5477  5482  5487  5492  5497  5501  5506  5511  5516 

5521  5525  5530  5535  5540  5545  5550  5554  5559  5564 

5569  6574  5578  5583  5588  5593  5598  5602  5607  6612 

5617  5622  5626  5631  5636  5641  5646  5650  5655  5660 

5665  5670  5674  5679  5684  5689  5694  5698  5703  5708 

5713  5718  5722  5727  5732  5737  5742  5746  5751  5756 

5761  5766  5770  5775  5780  5785  5789  5794  5799  5804 

5809  5813  5818  5823  5828  5832  5837  5842  5847  5852 

5856  5861  5866  5871  5875  5880  5885  5890  5895  5899 

910  95904  95909  95914  95918  95923  95928  95933  95938  95942  95947 

5952  5957  5961  5966  5971  5976  5980  5985  5990  5995 

5999  6004  6009  6014  6019  6023  6028  6033  6038  6042 

6047  6052  6057  6061  6066  6071  6076  6080  6085  6090 

6095  6099  6104  6109  6114  6118  6123  6128  6133  6137 

6142  6147  6152  6156  6161  6166  6171  6175  6180  6185 

6190  6194  6199  6204  6209  6213  6218  6223  6227  6232 

6237  6242  6246  6251  6256  6261  6265  6270  6275  6280 

6284  6289  6294  6298  6303  6308  6313  6317  6322  6327 

6332  6336  6341  6346  6350  6355  6360  6365  6369  6374 

96379  96384  96388  96393  96398  96402  96407  96412  96417  96421 

6426  6431  6435  6440  6445  6450  6454  6459  6464  6468 

6473  6478  6483  6487  6492  6497  6501  6506  6511  6515 

6520  6525  6530  6534  6539  6544  6548  6553  6558  6562 

6567  6572  6577  6581  6586  6591  6595  6600  6605  6609 

6614  6619  6624  6628  6633  6638  6642  6647  6652  6656 

6661  6666  6670  6675  6680  6685  6689  6694  6699  6703 

6708  6713  6717  6722  6727  6731  6736  6741  6745  6750 

6755  6759  6764  6769  6774  6778  6783  6788  6792  6797 

6802  6806  6811  6816  6820  6825  6830  6834  6839  6844 

96848  96853  96858  96862  96867  96872  96876  96881  96886  96890 

6895  6900  6904  6909  6914  6918  6923  6928  6932  6937 

6942  6946  6951  6956  6960  6965  6970  6974  6979  6984 

6988  6993  6997  7002  7007  7011  7016  7021  7025  7030 

7035  7039  7044  7049  7053  7058  7063  7067  7072  7077 

7081  7086  7090  7095  7100  7104  7109  7114  7118  7123 

7128  7132  7137  7142  7146  7151  7155  7160  7165  7169 

7174  7179  7183  7188  7192  7197  7202  7206  7211  7216 

7220  7225  7230  7234  7239  7243  7248  7253  7257  7262 

7267  7271  7276  7280  7285  7290  7294  7299  7304  7308 

97313  97317  97322  97327  97331  97336  97340  97345  97350  97354 

7359  7364  7368  7373  7377  7382  7387  7391  7396  7400 

7405  7410  7414  7419  7424  7428  7433  7437  7442  7447 

7451  7456  7460  7465  7470  7474  7479  7483  7488  7493 

7497  7502  7506  7511  7516  7520  7525  7529  7534  7539 

7543  7548  7552  7557  7562  7566  7571  7575  7580  7585 

7589  7594  7598  7603  7607  7612  7617  7621  7626  7630 

7635  7640  7644  7649  7653  7658  7663  7667  7672  7676 

7681  7085  7690  7695  7699  7704  7708  7713  7717  7722 

7727  7731  7736  7740  7745  7749  7754  7759  7763  7768 

97772  97777  97782  97786  97791  97795  97800  97804  97809  97813 


TABLE   IX.  —  LOGARITHMS    OF   NUMBERS. 


0123456789 


3 
4 
5 
6 
7 
8 
9 

960 

1 
2 
3 
4 
5 
6 
7 
8 
9 


980 

1 

2 
3 

4 
5 
6 

7 


97772  97777  97782  97786  97791  97795  97800  97804  97809  97813 

7818  7823  7827  7832  7836  7841  7845  7850  7855  7859 

7864  7868  7873  7877  7882  7886  7891  7896  7900  7905 

7909  7914  7918  7923  7928  7932  7937  7941  7946  7950 

7955  7959  7964  7968  7973  7978  7982  7987  7991  7996 

8000  8005  8009  8014  8019  8023  8028  8032  8037  8041 

8046  8050  8055  8059  8064  8068  8073  8078  8082  8087 

8091  8096  8100  8105  8109  8114  8118  8123  8127  8132 

8137  8141  8146  8150  8155  8159  8164  8168  8173  8177 

8182  8186  8191  8195  8200  8204  8209  8214  8218  8223 

98227  98232  98236  98241  98245  98250  98254  98259  98263  98268 

8272  8277  8281  8286  8290  8295  8299  8304  8308  8313 

8318  8322  8327  8331  8336  8340  8345  8349  8354  8358 

8363  8367  8372  8376  8381  8385  8390  8394  8399  8403 

8408  8412  8417  8421  8426  8430  8435  8439  8444  8448 

8453  8457  8462  8466  8471  8475  8480  8484  8489  8493 

8498  8502  8507  8511  8516  8520  8525  8529  8534  8538 

8543  8547  8552  8556  8561  8565  8570  8574  8579  8583 

8588  8592  8597  8601  8605  8610  8614  8619  8623  8628 

8632  8637  8641  8646  8650  8655  8659  8664  8668  8673 

98677  98682  98686  98691  98695  98700  98704  98709  98713  98717 

8722  8726  8731  8735  8740  8744  8749  8753  8758  8762 

8767  8771  8776  8780  8784  8789  8793  8798  8802  8807 

8811  8816  8820  8825  8829  8834  8838  8843  8847  8851 

8856  8860  8865  8869  8874  8878  8883  8887  8892  8896 

8900  8905  8909  8914  8918  8923  8927  8932  8936  8941 

8945  8949  8954  8958  8963  8967  8972  8976  8981  8985 

8989  8994  8998  9003  9007  9012  9016  9021  9025  9029 

9034  9038  9043  9047  9052  9056  9061  9065  9069  9074 

9078  9083  9087  9092  9096  9100  9105  9109  9114  9118 

99123  99127  99131  99136  99140  99145  99149  99154  99158  99162 

9167  9171  9176  9180  9185  9189  9193  9198  9202  9207 

9211  9216  9220  9224  9229  9233  9238  9242  9247  9251 

9255  9260  9264  9269  9273  9277  9282  9286  9291  9295 

9300  9304  9308  9313  9317  9322  9326  9330  9335  9339 

9344  9348  9352  9357  9361  9366  9370  9374  9379  9383 

9388  9392  9396  9401  9405  9410  9414  9419  9423  9427 

9432  9436  9441  9445  9449  9454  9458  9463  9467  9471 

9476  9480  9484  9489  9493  9498  9502  9506  9511  9515 

9520  9524  9528  9533  9537  9542  9546  9550  9555  9559  ! 

99564  99568  99572  99577  99581  99585  99590  99594  99599  99603 

9607  9612  9616  9621  9625  9629  9634  9638  9642  9647  , 

9651  9656  9660  9664  9669  9673  9677  9682  9686  9691  , 


1 
2 
3 
4 
5 
6 
7 
8 
9 

1000:  00000  00004  00009  00013  00017  00022  00026  00030  00035  00039 

"212 


9695  9699  9704  9708  9712  9717  9721  9726  9730  9734  ! 

9739  9743  9747  9752  9756  9760  9765  9769  9774  9778  | 

9782  9787  9791  9795  9800  9804  9808  9813  9817  9822  ; 

9826  9830  9835  9839  9843  9848  9852  9856  9861  9865 

9870  9874  9878  9883  9887  9891  9896  9900  9904  9909 

9913  9917  9922  9926  9930  9935  9939  9944  9948  9952 

9957  9961  9965  9970  9974  9978  9983  9987  9991  9996 


TABLE  X.— SINES  AND  COSINES. 


0° 

1 

0 

2° 

3° 

4° 

/ 

Sine  Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine 

Cosin 

Sine 

Cosin 

o 

00000 

One. 

01745 

99985 

03490 

99939 

05234 

99863 

06976 

99756 

60 

1 

U0029 

One. 

01774 

99984 

03519 

99938 

05263 

99861 

07005 

99754 

59 

2 

00058 

One. 

01803 

99984 

03548 

99937 

05292 

99860 

07034 

99752 

58 

3 

00087 

One. 

01832 

99983 

03577 

99936 

05321 

99858 

07063 

99750 

57 

4 

00116 

One. 

01862 

99983 

03606 

99935 

05350 

99857 

07092 

99748 

56 

5 

00145 

One. 

01891 

99982 

03635 

99934 

05379 

99855 

07121 

99746 

55 

6 

00175 

One. 

01920 

99982 

03664 

99933 

05408 

99854 

07150 

99744 

54 

7 

00204 

One. 

01949 

99981 

03693 

99932 

05437 

99852 

07179 

99742 

53 

8 

00233 

One. 

01978 

99980 

03723 

99931 

05466 

99851 

07208 

99740 

52 

9 

00262 

One. 

02007 

99980 

03752 

99930 

05495 

99849 

07237 

99738 

51 

iO 

00291 

One. 

02036 

99979 

03781 

99929 

05524 

99847 

07266 

99736 

50 

11 

00320 

99999 

02065 

99979 

OS810 

99927 

05553 

99846 

07295 

99734 

49 

12 

00349 

99999 

02094 

99978 

03839 

99926 

05582 

99844 

07324 

99731 

48 

13 

00378 

99999 

02123 

99977 

03868 

99925 

05611 

99842 

07353 

99729 

47 

14 

00407 

99999 

02152 

99977 

03897 

99924 

05640 

99841 

07382 

99727 

46 

15 

00436 

99999 

02181 

99976 

'03926 

99923 

05669 

99839 

07411 

99725 

45 

16 

00465 

99999 

02211 

99976 

03955 

99922 

05698 

99838 

07440 

99723 

44 

17 

00495 

99999 

02240 

99975 

03984 

99921 

05727 

99836 

07469 

99721 

43 

18 

00524 

99999 

02269 

99974 

04013 

99919 

05756 

99834 

07498 

99719 

42 

19 

00553 

99998 

02298 

99974 

04042 

99918 

05785 

99833 

07527 

99716 

41 

20 

00582 

99998 

02327 

99973 

04071 

99917 

05814 

99831 

07556 

99714 

40 

21 

00611 

99998 

02356 

99972 

04100 

99916 

05S44 

99829 

07585 

99712 

39 

22 

00640 

99998 

02385 

99972 

04129 

99915 

05873 

99827 

07614 

99710 

38 

23 

00669 

99998 

02414 

99971 

04159 

99913 

05902 

99826 

07643 

99708 

37 

24 

00698 

99998 

02443 

99970 

04188 

99912 

05931 

99824 

07672 

99705 

36 

25 

00727 

99997 

02472 

99969 

04217 

99911 

05960 

99822 

07701 

99703 

35 

26 

00756 

99997 

02501 

99969 

04246 

99910 

05989 

99821 

07730 

99701 

34 

27 

00785 

99997 

02530 

99968 

04275 

99909 

06018 

99819 

07759 

99699 

33 

28 

00814 

99997 

02560 

99967 

04304 

99907 

06047 

99817 

07788 

99696 

32 

29 

00844 

99996 

02589 

99966 

04333 

99906 

06076 

99815 

07817 

99694 

31 

30 

00873 

99996 

02618 

99966 

04362 

99905 

06105 

99813 

07846 

99692 

30 

31 

00902 

99996 

02647 

99965 

04391 

99904 

06134 

99812 

07875 

99689 

29 

32 

00931 

99996 

02676 

99964 

04420 

99902 

06163 

99810 

07904 

99687 

28 

33 

00960 

99995 

02705 

99963 

04449 

99901 

06192 

99808 

07933 

99685 

27 

34 

00989 

99995 

02734 

99963 

04478 

99900 

06221 

99806 

07962 

99683 

26 

35 

01018 

99995 

02763 

99962 

04507 

99898 

06250 

99804 

07991 

99680 

85 

36 

01047 

99995 

02792 

99961 

04536 

99897 

06279 

99803 

08020 

99678 

24 

37 

01076 

99994 

02821 

99960 

04565 

99896 

06308 

99801 

08049 

99676 

23 

38 

01105 

99994 

02850 

99959 

04594 

99894 

06337 

99799 

08078 

99673 

22 

39 

01134 

99994 

02879 

99959 

04623 

99893 

06366 

99797 

08107 

99671 

21 

40 

01164 

99993 

02908 

99958 

04653 

99892 

06395 

99795 

08136 

99668 

20 

41 

01193 

99993 

02938 

99957 

04682 

99890 

06424 

99793 

08165 

99666 

19 

42 

01222 

99993 

02967 

99956 

04711 

99889 

06453 

99792 

08194 

99664 

18 

43 

01251 

99992 

02996 

99955 

04740 

99888 

06482 

99790 

08223 

99661 

17 

44 

01280 

99992 

03025 

99954 

04769 

99886 

06511 

99788 

08252 

99659 

16 

45 

01309 

99991 

03054 

99953 

04798 

99885 

06540 

99786 

08281 

99657 

15 

46 

01338 

99991 

03083 

99952 

04827 

99883 

06569 

99784 

08310 

99654 

14 

47 

01367 

99991 

03112 

99952 

04856 

99882 

06598 

99782 

08339 

99652 

13 

48 

01396 

99990 

03141 

99951 

04885 

99881 

06627 

99780 

08368 

99649 

12 

49 

01425 

99990 

03170 

99950 

04914 

99879 

06656 

99778 

08397 

99647 

11 

50 

01454 

99989 

03199 

99949 

04943 

99878 

06685 

99776 

08426 

99644 

10 

51 

01483 

99989 

03228 

99948 

04972 

99876 

06714 

99774 

08455 

99642 

9 

52 

01513 

99989 

99947 

05001 

99875 

06743 

99772 

08484 

99639 

8 

53 

01542 

99988 

03286 

99946 

05030 

39873 

06773 

99770 

08513 

99637 

7 

54 

01571 

99988 

03316 

99945 

05059 

99872 

06802 

99768 

08542 

99635 

6 

55 

0160G 

99987 

03345 

99944 

05088 

99870 

06831 

99766 

08571 

99632 

5 

56 

01629 

99987 

03374 

99943 

05117 

99869 

06860 

99764 

08600 

99630 

4 

57 

01658 

99986 

03403 

99942 

05146 

99867 

06889 

99762 

08629 

99627 

3 

58 

01687 

99986 

03432 

99941 

05175 

99866 

06918 

99760 

08658 

99625 

2 

59 

01716 

99985 

03461 

99940 

05205 

99864 

06947 

99758 

0868V 

99622 

1 

60 

01745 

99985 

03490 

99939 

05234 

99863 

06976 

99756 

08716 

99619 

J) 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 

/ 

89°       88° 

87° 

86' 

85° 

213 


TABLE  X.— SINES  AND  COSINES. 


37 


Sine 
0871 
08745 
08774 


13975 
14004 
14033 
14061 
14090 
14119 
14148 
14177 
14205 


99607 
99604 
99602 
99599 


10771 
10800 
10829 
10858 
10887 
10916 
10945 
10973 
11002 
11031 


12504 
12533 
12562 
12591 


15959 
15988 
16017 
16046 
16074 
16103 
16132 
16160 
16189 
16218 


99586 
99583 
99580 
9957S 
99575 
99572 
99570 
99567 


12649 

12678 
12706 
12735 
12764 


99178 
99175 
99171 
99167 
99163 
99160 
99156 
99152 
99148 
99144 


14522 
14551 
14580 
14608 
14637 
14666 


99380 
99377 
99374 
99370 
99367 
99364 
99360 
99357 


16361 
16390 
16419 
16447 
16476 
16505 


99551 
99548 
99545 
99542 
99540 


11349 
11378 
11407 
11436 
11465 
11494 
11523 
11552 
11580 
11609 


99354 
99351 
99347 
99344 
99341 
99337 
99334 


13081 
13110 
13139 
13168 
13197 


16533 
16562 
16591 
16620 
16648 
16677 
16706 
16734 
16763 
16792 


l  1985 
14954 
14982 
15011 
15040 
15069 


13254 
13283 
13312 
13341 


13370 
13399 
13427 
1345G 
13485 
13514 
13543 
13572 
13600 
13629 


16820 

16849 
16878 
16906 


15097 
15126 
15155 
15184 
15212 
15241 
15270 
15299 
15327 
15350 


99506 
99503 
99500 
99497 
99494 
99491 
99488 
99485 


11927 
11956 
11985 
12014 
12043 
12071 
12100 
12129 
12158 
12187 
Cosin 


13658 
136S7 
13716 
13744 
13773 
13802 
13831 
13860 


15385 
15414 
15442 
15471 
15500 
15529 
15557 
15586 
15615 
15643 
Cosin 


17107 
17136 
17164 
17193 
17222 
17250 
17279 
17308 
17336 
17365 


99063 
99059 
99055 
99051 
99047 
99043 


98511 
98506 
98501 
98496 
98491 


84° 


214 


TABLE  X.-SINES  AND  COSINES. 


1 

10° 

11° 

12° 

13 

14° 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine  Cosin 

0 

17365 

98481 

'19081 

98163 

"20791 

97815 

22495 

97437 

24192 

97030 

60 

1 

17393 

98476 

19109 

98157 

20820 

97809 

22523 

97430 

24220 

97023 

59 

2 

17422 

98471 

19138 

98152 

20848 

97803 

22552 

97424 

2424'J 

97015 

58 

3 

17451 

98466 

19167 

98146 

20877 

97797 

22580 

97417 

24277 

97008 

57 

4 

17479 

98461 

19195 

98140 

20905 

97791 

22608 

97411 

24305 

97001 

56 

5 

17508 

98455 

19224 

9S135 

20933 

97784 

22637 

97404 

24:333 

96994 

55 

6 

17537 

98450 

19252 

98129 

20962 

97778 

22665 

07398 

24362 

96987 

54 

7 

17565 

98445 

19281 

98124 

20990 

97772 

22693 

97391 

24390 

96980  53 

8 

17594 

98440 

19309 

98118 

21019 

97766 

22722 

97384 

24418 

96973)  52 

9 

17623 

98435 

19338 

98112 

21047 

97760 

22750 

97378 

24446 

96966  51 

10 

17651 

98430 

19366 

98107 

21076 

97754 

22778 

97371 

24474 

96959 

50 

11 

17680 

98425 

19395 

98101 

21104 

97748 

22807 

97365 

24503 

96952 

49 

12 

17708 

98420 

19423 

98096 

21132 

97742 

22835 

97358 

24531 

96945,  48 

13 

17737 

98414 

19452 

98C90 

21161 

97735 

22863 

97351 

24559 

96937 

47 

14 

17766 

98409 

19481 

98084 

21189 

97729 

22892 

97345 

24587 

96930 

46 

15  17794 

98404 

19509 

98079 

21218 

97723 

22020 

97338 

24615 

96923!  45 

16  17823 

98399 

19538 

98073 

21246 

97717 

22948 

97331 

24644 

96916  44 

17 

17852 

98394 

19566 

98067 

21275 

97711 

22977 

97325 

24672 

96909  43 

18 

17880 

98389 

19595 

98061 

21303 

97705 

23005 

97318 

24700 

96902  42 

19 

17909 

98383 

19623 

98056 

21331 

97698 

23033 

97311 

24728 

968941  41 

20 

17937 

98378 

19652 

98050 

21360 

97692 

23062 

97304 

24756 

968871  40 

21 

17966 

98373 

19680 

98044 

21388 

97686 

23090 

97298 

24784 

96880!39 

22 

17995 

98368 

19709 

98039 

21417 

97680 

23118 

97291 

24813 

968731  38 

23 

18023 

98362 

19737 

93033 

21445 

97673 

23146 

97284 

24841 

9686(5  !  37 

24 

18052 

98357 

19766 

98027 

21474 

97667 

23175 

97278 

24869 

96858  36 

25 

18081 

98352 

19794 

98021 

21502 

97661 

23203 

97271 

24897 

96851  i  35 

26 

18109 

98347 

19823 

98016 

21530 

97655 

23231 

97264 

2-4925 

96844134 

27 

18138 

98341 

19851 

98010 

21559 

97648 

23260 

97257 

24954 

96837  33 

28 

18166 

98336 

19880 

93004 

21587 

97642 

23288 

97251 

24982 

96829 

32 

29 

18195 

98331 

19908 

97998 

21616 

97636 

23316 

97244 

25010 

96822 

31 

30 

18224 

98325 

19937 

97992 

21644 

97630 

23345 

97237 

25038 

96815 

30 

31 

18252 

98320 

19965 

97987 

21672 

97623 

23373 

97230 

25066 

96807 

29 

32 

18281 

98315 

19994 

97981 

21701 

97617 

23401 

97223 

25094 

96800 

28 

33 

18309 

98310 

20022 

97975 

21729 

97611 

23423 

97217 

25122 

96793 

27 

34 

18338 

98304 

20051 

97969 

21758 

97604 

23458 

97210 

25151 

96786  26 

35 

18367 

98299 

20079 

97963 

21786 

97598 

23486 

97203 

25179 

96778'  25 

36 

18395 

98294 

20108 

97958 

21814 

97592 

23514 

97196 

25207 

96771 

24 

37 

18424 

98288 

20136 

97952 

21843 

97585 

23542 

97189 

25235 

96764 

23 

38 

18452 

98283 

20165 

97946 

21871 

97'579 

23571 

97182 

25263 

96756 

22 

39 

18481 

98277 

20193 

97940 

21899 

97573 

23599 

97176 

25291 

96749 

21 

40 

18509 

98272 

20222 

97934 

21928 

97566 

23627 

97169 

25320 

96742 

20 

41 

18538 

98267 

20250 

97928 

21956 

97560 

23656 

97162 

25348 

96734 

19 

42 

18567 

98261 

20279 

97922 

97553 

23684 

97155 

25376 

96727 

18 

43 

18595 

98256 

20307 

97916 

22013 

97547 

23712 

97148 

25404 

96719 

17 

44 

18624 

98250 

20336 

97910 

22041 

97541 

23740 

97141 

25432 

96712!  16 

45 

18652 

98245 

20364 

97905 

22070 

97534 

23769 

97134 

25460 

9G705  15 

46 

18681 

98240 

20393 

97899 

22098 

97528 

23797 

97127 

25488 

96697  14 

47 

18710 

98234 

20421 

97893 

22126 

97521 

23825 

97120 

25516 

96690  13 

48 

18738 

98229 

20450 

97887 

22155 

97515 

23853 

97113 

25545 

96682  12 

49 

18767 

98223 

20478 

97881 

22183 

97508 

23882 

97106 

25573 

96675  11 

50 

18795 

98218 

20507 

97875 

22212 

97502 

23910 

97100 

25601 

96667  10 

51 

18824 

98212 

20535 

97869 

22240 

97496 

23938 

97093 

25629 

96660 

9 

52 

18852 

98207 

20563 

97863 

22268 

97489 

23966 

97086 

25657 

96653,  8 

53 

18881 

98201 

20592 

97857 

22297 

97483 

23995 

97079 

25685 

96645  7 

54 

18910 

98196 

20620 

97851 

22325 

97476 

24023 

97072 

25713 

966:38  6 

55 

18938 

98190 

20649 

97845 

22353 

97470 

24051 

97065 

25741 

96630  5 

56 

18967 

98185 

20677 

97839 

22382 

97463 

24079 

97058 

25769 

96623  4 

57 

18995 

98179 

20706 

97833 

22410 

37457 

24108 

97051 

25798 

96615  3 

58 

19024 

98174 

207:34 

97827 

22438 

97450 

24136 

97044 

25826 

96608  2 

59 

19052 

98168 

20763 

9782^ 

22467 

97444 

24164 

97037 

25854 

96600  1 

GO 

19081 

98163 

20791 

9781? 

22495 

97437 

24192 

97030 

25882 

96593  0 

t 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

f 

79° 

78° 

„. 

76 

0 

75 

0 

215 


TABLE  X.-SINES  AND  COSINES. 


15°       16°       17°       18°       19° 

Sine  Cosin  Sine  Cosin  Sine 

Cosin  Sine 

Cosin  Sine  Cosin 

0 

25882  96593  27564  96126  29237 

95630   30902 

95106   32557  .94552 

60 

1 

25910  96585   27592  96118  29265 

95622   30929 

95O7  32584  94542 

59 

2 

25938  96578  27620  96110  29293 

95613  30957 

95088  32612  94533 

58 

3 

25966  96570  27648  96102  29321 

95605  30985 

95079  32639  94523 

57 

4 

25994  96562   27676  96094   29348 

95596   31012 

95070  32667  94514 

56 

5 

26022  96555  27704  96086  29376 

95588  31040 

95061   32694  94504 

55 

6 

26050  96547  27731  96078  29404 

95579  31068 

95052  32722  94495 

54 

7 

26079  96540  27759  96070  29432 

95571   31095 

95043  32749  94485 

53 

8 

26107  96532  27787  96062  29460 

95562   31123 

95033   32777  94476 

52 

9 

26135  96524   27815  96054  29487 

95554  31151 

95024  32804  94466 

51 

10 

26163  96517  27843  96046  29515 

95545  31178 

95015  32832  94457 

50 

11 

26191  96509  27871  96037  29543 

95536  31206 

95006  32859  94447 

49 

12 

26219  96502  27899  96029  29571 

95528   31233 

94997  32887  94438 

48 

13 

26247  96494   27927  96021   29599 

95519   31261 

94988  32914  94428 

47 

14 

26275  96486  27955  96013  29626 

95511   31289 

94979  32942  94418 

46 

15 

26303  96479.  27983  96005  29654 

95502   31316 

94970  32969  94409 

45 

16 

26331  96471  28011  95997  29682 

95493  31344 

94961  32997  94399 

44 

17 

26359  96463  28039  95989  29710 

95485   31372 

94952   33024  94390 

43 

18 

2638?  96456  28067  95981   29737 

95476  31399 

94943  33051  94380 

42 

19 

26415  96448  28095  95972  29765 

95467  31427 

94933  33079  94370 

41 

20 

26443  96440  28123  95964  29793 

95459  31454 

94924  33106  94361 

40 

21 

26471  96433  28150  95956  29821 

95450  31482 

94915  33134  94351 

39 

22 

26500  96425  28178  95948  29849 

95441,  31510 

94906  33161  94342 

38 

23 

26528  96417  28206  95940  29876 

95433,  31537 

94897  33189  94332 

37 

24 

26556  96410  28234  95931   29904 

95424  31565 

94888   33216  94322 

36 

25 

26584  96402  28262  95923  29932 

95415  31593 

94878   33244  94313 

35 

26 

26612  96394  28290  95915  29960 

95407  31620 

94869  33271  94303 

34 

27 

26640  96386  28318  95907  29987 

95398  31648 

94860  33298  94293 

33 

28 

26668  96379  28346  95898  30015 

95389  31675 

94851   33326  94284 

32 

29 

26696  96371   28374  95890  30043 

95380  31703 

94842  33353  94274 

31 

30 

26724  96363  28402  95882  30071 

95372  31730 

94832  33381  94264 

30 

31 

26752  96355  28429  95874  30098 

95363  31758 

94823  33408  94254 

29 

32 

26780  9634?  28457  95865  30126 

95354  31786 

94814  33436  94245 

28 

33 

26808  96340  28485  95857  30154 

95345  31813 

94805   33463  94235 

27 

34 

26836  96332  28513  95849  30182 

95337  31841 

94795  33490  94225 

26 

35 

26864  96324  28541  95841   30209 

95328  31868 

94786  33518  94215 

25 

36 

26892  96316  28569  95832  30237 

95319  31896 

94777  33545  94206 

24 

37 

26920  96308  28597  95824  30265 

95310  31923 

94768   33573  94196 

23 

38 

26948  96301   28625  95816  30292 

95301   31951 

94758  33600  94186 

22 

39 

26976  96293   28652  95807  '30320 

95293   31979 

94749   33627  94176 

21 

40 

27004  96285  28680  95799  30348 

95284  32006 

94740  33655  94167 

20 

41 

27032  96277  28708  95791  30376 

95275  32034 

94730  33682  94157 

19 

42 

27060  96269  28736  95782  30403 

95266  32061 

94721   33710  94147 

18 

43 

27088  96261   28764  95774  30131 

95257   32089 

94712  38737  94137 

17 

44 

27116  96253  28792  95766  30459 

95248  32116 

94702  33764  94127 

16 

45 

27144  96246  28820  95757  30486 

95240  32144 

94693  33792  94118 

15 

46 

27172  96238  28847  95749  30514 

95231   32171 

94684  33819  94108 

14 

47 

27200  96230  28875  95740  30542 

95222   32199 

94674  33846  94098 

13 

48 

27228  96222  28903  95732  30570 

95213  32227 

94665  33874  94088 

12 

49 

27256  96214  28931  95724  30597 

95204   32254 

94656   33901  94078 

11 

50 

27284  96206  28959  95715  30625 

95195-  32283 

94641)  33929  94068 

10 

51 

27312  96198  28987  95707  30653 

95186  32309 

94637  33956  94058 

9 

52 

27340  96190  29015  95698  30680 

9517?  32337 

94G27   33983  94049 

8 

53 

27368  96182  29042  95690  30708 

95168  32364 

94618  34011  94039 

7 

54 

27396  96174   29070  95681   30736 

95159  32392 

94609  34038  94029 

6 

55 

27424  96166   29098  95673  30763 

95150  32419 

94599  34065  94019 

5 

56 

27452  96158   29126  95664  30791 

95142   32447 

94590  34093  94009 

4 

57 

27480  96150  29154  95656  30819 

95133  32474 

93580  34120  93999 

3 

58 

27508  96142  29182  95647  30846 

95124  32502 

94571   34147  93989 

2 

59 

27536  96134   29209  95639  30874 

95115   32529 

94561  34175  93979 

1 

60 

27564  96126   29237  95630   30902 

95106  .  32557  , 

94552  34202  93969 

0 

/ 

Cosin  Sine  Cosin  Sine  Cosin 

Sine  Cosin 

Sine  Cosin  Sine 

/ 

74°       73°       72°       71°       70° 

216 


TABLE  X.— SINES  AND  COSINES. 


20' 

21 

o 

a 

5<» 

23 

0 

24 

' 

Sine 

Cosin 

Sine  Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine  Cosin 

~o 

34202 

93969 

35837 

93358 

37461 

92718 

39073 

92050 

40674 

91355 

60 

1 

34229 

93959 

35864 

93348 

37488 

92707 

39100 

92039 

40700 

91343 

59 

2 

34257 

93949 

35891 

93337 

37515 

92697 

39127 

92028 

40727 

91331  1  58 

3 

34284 

93939 

35918 

93327 

37542 

92686 

39153 

92016 

40753 

91319'  57 

4 

34311 

93929 

35945 

93316 

37569 

92675 

39180 

92005 

40780 

91307 

56 

5 

34339 

93919 

35973 

93306 

37595 

92664 

39207 

91994 

40806 

91295 

55 

6 

34366 

93909 

36000 

93295 

37622 

92653 

39234 

91982 

40833 

91283 

54 

7 

34393 

93899 

36027 

93285 

37649 

92642 

39260 

91971 

40860 

91272 

53 

8 

34421 

93889 

36054 

93274 

37676 

92631 

39281; 

91959 

40886 

91260 

52 

9 

34448 

93879 

36081 

93264 

37703 

92620  1 

39314 

91948 

40913 

91248 

51 

10 

34475 

93869 

36108 

93253 

37730 

92609 

393-41 

91936 

40939 

91236 

50 

11 

34503 

93859 

36135 

93243 

37757 

92598 

39367 

91925 

40966 

91224 

49 

12 

34530 

93840 

361G2 

93232 

37784 

92587 

39394 

91914 

40992 

91212 

48 

13 

34557 

93839 

36190 

93222 

37811 

92576 

39421 

91902 

41019 

91200 

47 

14 

34584 

93829 

36217 

93211 

37838 

92565 

39448 

91891 

41045 

91188 

46 

15 

34612 

93819 

36244 

93201 

37865 

92554 

39474 

91879 

41072 

91176 

45 

16 

34G39 

93809 

36271 

93100 

37892 

92543 

39501 

91868 

41098 

91164 

44 

17 

34G66 

93799 

36298 

93180 

37919 

92532 

39528 

91856 

41125 

91152  43 

18 

84094 

93789 

36325 

931G3 

37946 

92521 

39555 

91845 

41151 

91140  J£ 

19 

34721 

93779 

36352 

93159 

37973 

92510 

39581 

91833 

41178 

91128 

41 

20 

34748 

93769 

36379 

93148 

37999 

.92499 

39608 

91822 

41204 

91116 

40 

21 

34775 

93750 

36406 

93137 

38026 

92488 

39635 

91810 

41231 

91104 

39 

22 

34803 

93748 

36434 

93127 

38053 

92477 

39661 

91799 

41257 

91092 

38 

23 

34830 

93738 

36461 

9311G 

38080 

92466 

39688 

91787 

41284 

91080 

37 

24 

34857 

93728 

36488 

93106 

38107 

92455 

39715 

91775 

41310 

91068 

36 

25 

34884 

93718 

36515 

93095 

38134 

92444 

39741 

91764 

41337 

91056 

35 

26 

34912 

93708 

36542 

93084 

38161 

92432 

39768 

91752 

41363 

91044 

34 

27 

34939 

93698 

3G5G9 

93074 

38188 

92421 

39795 

91741 

41390 

91032 

33 

23 

34966 

93688 

36596 

93063 

38215 

92410 

39822 

91729 

41416 

91020 

32 

29 

34993 

93677 

3GG23 

93052 

£8241 

92399 

39846 

91718 

41443 

91008 

31 

30 

35021 

93667 

36650 

93042 

38268 

92388 

39875 

91706 

41469 

90996 

30 

31 

35048 

93657 

36677 

93031 

38295 

92377 

39902 

91694 

41496 

90984 

29 

32 

35075 

93647 

36704 

93020 

38322 

92366 

39928 

91C88 

41522 

9097'2 

28 

33 

35102 

93637 

36731 

93010 

38349 

92355 

39955 

91671 

41549 

90960 

27 

34 

35130 

93626 

36758 

92999 

38376 

92343 

39982 

91660 

41575 

90948 

26 

35 

35157 

93616 

36785 

92988 

38403 

92332 

40008 

91648 

41602 

90936 

25 

36 

35184 

93606 

36812 

92978 

38430 

92321 

40035 

91636 

41628 

90924 

24 

37 

35211 

93596 

36839 

92967 

38456 

92310 

40062 

91625 

41655 

90911 

23 

38 

35239 

93585 

36867 

92956 

38483 

92299 

40088 

91613 

41681 

90899 

22 

39 

35266 

93575 

36894, 

92945 

38510 

92287 

40115 

91G01' 

41707 

90887 

21 

40 

35293 

93565 

36921 

92935 

38537 

92276 

40141 

91590 

41734 

90875 

20 

41 

35320 

93555 

36948 

92924 

38564 

92265 

40168 

91578 

41760 

90863 

19 

42 

35347 

93544 

36975 

92913 

38591 

92254 

40195 

91566 

41787 

90851 

18 

43 

35375 

93534 

37002 

92902 

38617 

92243 

40221 

91555 

41813 

90839 

17 

44 

35402 

93524 

37029 

92892 

38644 

92231 

40248 

91543 

41840 

90826 

16 

45 

35429 

93514 

37056 

92881 

38671 

92220 

40275 

91531 

41866 

90814 

15 

46 

35456 

93503 

37083 

92870 

38698 

92209 

40301 

91519 

41892 

90802  14 

47 

35484 

93493 

37110 

92859 

38725 

92198 

40328 

91508 

41919 

90790  13 

48 

35511 

93483 

37137 

92849 

38752 

92186 

40355 

91496 

41945 

90778  12 

49 

35538 

93472 

37164 

92838 

38778 

92175 

40381 

91484 

41972 

90766  11 

50 

35565 

93462 

37191 

92827 

38805 

92164 

40408 

91472 

41998 

90753 

10 

51 

35592 

93452 

37218 

92816 

38832 

92152 

40434 

91461 

42024 

90741 

9 

52 

35619 

93441 

37245 

92805 

38859 

92141 

40461 

91449 

42051 

90729 

8 

53 

35647 

93431 

37272 

92794 

38886 

92130 

40488 

91437 

42077 

90717 

7 

54 

35674 

93420 

37299 

92784 

38912 

92119 

40514 

91425 

42104 

90704 

6 

55 

35701 

93410 

37326 

92773 

38939 

92107 

40541 

91414 

42130 

90692 

5 

56 

35728 

93400 

37353 

92762 

38966 

92096 

40567 

91402 

42156 

90680 

4 

57 

35755 

93389 

37380 

92751 

38993 

92085 

40594 

91390 

42183 

90668 

3 

58 

35782 

9.3379 

37407 

92740 

39020 

92073 

40621 

91378 

42209 

90655 

2 

59 

35810 

93368 

37434 

92729 

39046 

92062 

40647 

91366 

42235 

90643 

1 

60 

35837 

93358 

37461 

92718 

39073 

92050 

40674 

91355 

42262 

90631 

0 

9 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

f 

69°       68° 

67° 

66° 

65 

o 

217 


TABLE  X.— SINES  AND  COSINES. 


Sine  Cosin 

48481  "87462 
48506  87448 
48532  87434 
48557  87421 
48583  8740( 


44124 
44151 
44177 
44203 
44229 
44255 
44281 
44307 
44333 
44359 


45684 
45710 
45736 
45762 
45787 
45813 


48761 

48786 
48811 
48837 


90483 
90470 
90458 
90446 
90433 
90421 
90408 


42762 

42788 

42815 
42841 


44385 
44411 
44437 
44464 
44490 
44516 
44542 
44568 
44594 
44620 


43077 
43104 
43130 
43156 


44646 
44672 
44698 
44724 
44750 
44776 
44802 


47741 
47767 
47793 
47818 
47844 
47869 
47895 
47920 
47946 
47971 


43418 
43445 
43471 
43497 


45166 
45192 
45218 
45243 
45269 
45295 
45321 
45347 
45373 


89219 
89206 
89193 
89180 
89167 
8915S 
89140 
89127 
89114 
89101 
Sine 


43706 
43733 
43759 
43785 
43811 
43837 


89943 
89930 
89918 
89905 
89892 
89879 


49924 
49950 
49975 
50000 


46921  88308 
46947  88295 
Cosin  Sine 


218 


TABLE  X.-SINES  AND  COSINES. 


30° 

31 

o 

32°       33° 

34° 

f 

/ 

Sine  Cosin  ' 

Sine  Cosin 

Sine  Cosin 

Sine 

Cosin 

Sine  Cosin 

~o 

~50000 

86G03 

^51504' 

85717 

52992 

84805 

54464 

83867 

'55919  82904 

60 

1 

50025 

86588 

51529 

85702 

53017 

84789 

54488 

83851 

55943 

82887 

5S 

2 

50050 

86573 

51554 

85687 

53041 

84774 

54513 

83835 

55968 

82871 

58 

3 

50076 

86559 

51579 

85672 

53066 

84759 

54537 

83819 

55992 

82855 

57 

4 

50101 

86544 

51604 

85657 

53091 

84743 

54561 

83804 

56016 

82839 

56 

5 

50126 

86530 

51628 

85G42 

53115 

84728 

54586 

83788 

56040 

82822 

55 

6 

50151 

86515 

51653 

85627 

53140 

84712 

54610 

83772 

56064 

82806 

54 

7 

50176 

86501 

51678 

85612 

53164 

84697 

54635 

83756 

56088 

82790 

53 

8 

50201 

86486 

51703 

85597 

53189 

84681 

54659 

83740 

56112 

82773 

52 

9 

50227 

86471 

51728, 

85582 

53214 

84666 

54683 

83724 

56136 

82757 

51 

10 

50252 

86457 

51753 

85567 

53238 

84650 

54708 

837U8 

56160 

82741 

50 

11 

50277 

86442 

51778 

85551 

53263 

84635 

54732 

83692 

56184 

82724 

4C 

12 

50302 

86427 

51803 

85536 

53288 

84619 

54756 

83676 

56208 

82708 

48 

13 

50327 

86413 

51828 

85521 

53312 

84604 

54781 

83660 

56232 

82692 

47 

14 

50352 

86398 

51852 

85506 

53337 

84588 

54805 

83645 

56256 

82675 

46 

15 

50377 

86384 

51877 

85491 

53361 

84573 

54829 

83629 

56280 

82659 

45 

16 

50403 

86369 

51902 

85476 

53386 

84557 

54854 

83613 

56305 

82643 

44 

17 

50428 

86354 

51927 

85461 

53411 

84542 

54878 

83597 

56329 

82626 

43 

18 

50453 

86340 

51952 

85446 

53435 

84526 

54902 

83581 

56353 

82610 

42 

19 

50478 

86325 

51977 

85431 

534.60 

84511 

54927 

83565 

56377 

82593 

41 

20 

50503 

86310 

52002 

85416 

53484 

84495 

54951 

83549 

56401 

82577 

40 

21 

50528 

86295 

52026 

85401 

53509 

84480 

54975 

83533 

56425 

82561 

89 

22 

50553 

86281 

52051 

85385 

53534 

84464 

54999 

83517 

56449 

82544 

88 

23 

50578 

86266 

52076 

85370 

53558 

84448 

55024 

83501 

56473 

82528  37 

24 

50C03 

86251 

52101 

85355 

53583 

84433 

55048 

83485 

56497 

S2511 

36 

25 

50628 

86237 

52126 

85340 

53607 

84417 

55072 

83469 

56521 

82495 

35 

26 

50654 

86222 

52151 

85325 

53632 

84402 

55097 

83453 

56545 

82478 

34 

27 

50679 

86207 

52175 

85310 

53656 

84386 

55121 

83437 

56569 

82462 

33 

28 

50704 

86192 

52200 

85294 

53681 

84370 

55145 

83421 

56593 

82446 

32 

29 

50729 

86178 

52225 

85279 

53705 

84355 

55169 

83405 

56617 

82429 

31 

30 

50754 

86163 

52250 

85264 

53730 

84339 

55194 

83389 

56641 

82413 

30 

31 

50779 

86148 

52275 

85249 

53754 

84324 

55218 

83373 

56665 

82396 

29 

32 

50804 

86133 

52299 

85234 

53779 

84308 

55242 

83356 

56689 

82380 

28 

33 

50829 

86119 

52324 

85218 

53804 

84292 

55266 

83340 

56713 

82363 

27 

34 

50854 

86104 

52349 

85203 

53828 

84277 

55291 

83324 

56736 

82347 

26 

35 

50879 

86089 

52374 

85188 

53853 

84261 

55315 

83308 

56760 

82330 

25 

36 

50904 

86074 

52399 

85173 

53877 

84245 

55339 

83292 

56784 

82314 

24 

37 

50929 

86059 

52423 

85157 

53902 

84230 

55363 

83276 

56808 

82297 

23 

38 

50954 

86045 

52448 

85142 

53926 

84214 

55388 

83260 

56832 

82281 

22 

39 

50979 

86030 

52473 

85127 

53951 

84198 

55412 

83244 

56856 

82264 

21 

40 

51004 

86015 

52498 

85112 

53975 

84182 

55436 

83228 

56880 

82248 

SO 

41 

51029 

86000 

52522 

85096 

54000 

84167 

55460 

83212 

56904 

82231 

19 

42 

51054 

85985 

52547 

85081 

54024 

84151 

55484 

83195 

56928 

82214 

18 

43 

51079 

85970 

52572 

85066 

54049 

84135 

55509 

83179 

56952 

82198 

17 

44 

51104 

85956 

52597 

85051 

54073 

84120 

55533 

83163 

56976 

82181 

16 

45 

51129 

85941 

52621 

85035 

54097 

84104 

55557 

83147 

57000 

82165 

15 

46 

51154 

85926 

52646 

85020 

54122 

84088 

55581 

83131 

57024 

82148 

14 

47 

51179 

85911 

52671 

85005 

54146 

84072 

55605 

83115 

57047 

82132 

13 

48 

51204 

85896 

52696 

'84989 

54171 

84057 

55630 

83098 

57071 

82115 

12 

49 

51229 

85881 

52720 

84974 

54195 

84041 

55654 

83082 

57095 

82098 

11 

50 

51254 

85866 

52745 

84959 

54220 

84025 

55678 

83066 

57119 

82082 

10 

51 

51279 

85851 

52770 

84943 

54244 

34009 

55702 

83050 

57143 

32065 

9 

52 

51304 

85836 

52794 

84928 

54269 

83994 

55726 

83034 

57167 

82048 

8 

53 

51329 

85821 

52819 

84913 

54293 

83978 

55750 

83017 

57191 

82032 

7 

54 

51354 

85806 

52844 

84897 

54317 

83962 

55775 

83001 

57215 

82015 

6 

55 

51379 

85792 

52869 

84882 

54342 

83946 

55799 

82985 

57238 

81999 

5 

56 

51404 

85777 

52893 

84866 

54366 

83930 

55823 

82969 

57262 

81982 

4 

57 

51429 

85762 

52918 

84851 

54391 

83915 

55847 

82953 

57286 

81965 

3 

58 

51454 

85747 

52943 

84836 

54415 

83899 

55871 

82936 

57310 

81949 

2 

59 

51479 

85732 

52967 

84820 

54440 

83883 

55895 

82920 

57334 

81932 

1 

60 

51504 

85717 

52992 

84805 

54464 

83867 

55919 

82904 

57358 

81915 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin  |  Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 

59° 

58° 

57° 

56°       55° 

219 


TABLE  X.-SINES  AND  COSINES. 


35 

0 

36° 

37 

38° 

39° 

Sine  Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine 

Cosin 

Sine 

Cosin 

f 

9 

57358 

81915 

^8779 

80902 

60182 

79864 

61566 

78801 

62932 

77715 

tiO 

u. 

57381 

81899 

58802 

80885 

60205 

79846 

61589 

78783 

62955 

77696 

59 

2 

57405 

81882 

58826 

80867 

60228 

79829 

61612 

78765 

62977 

77678 

58 

3 

57429 

81865 

58849 

80850 

60251 

79811 

61635 

78747 

63000 

77660 

57 

4 

57453 

81848 

58873 

80833 

60274 

79793 

61658 

78729 

63022 

77641 

56 

5 

57477 

81832 

58896 

80816 

60298 

79776 

61681 

78711 

63045 

77623 

55 

6 

57501 

81815 

58920 

80799 

60321 

79758 

61704 

78694 

63068 

77605 

54 

7 

57524 

81798 

58943 

80782 

60344 

79741 

61726 

78676 

63090 

77586 

53 

8 

57548 

81782 

58967 

80765 

60367 

79723 

61749 

78658 

63113 

77568 

52 

9 

57572 

81765 

58990 

80748 

60390 

79706 

61772 

78640 

63135 

77550 

51 

10 

57596 

81748 

59014 

80730 

60414 

79688 

61795 

78622 

63158 

77531 

50 

11 

57619 

81731 

59037 

80713 

60437 

79671 

61818 

78604 

63180 

'77513 

40 

12 

57643 

81714 

59061 

80696 

60460 

79653 

61841 

78586 

63203 

77494 

48 

13 

57667 

81698 

59084 

80679 

60483 

79635 

C1864 

78568 

63225 

77476 

47 

14 

57691 

81681 

59108 

80662 

60506 

79618 

61887 

78550 

63248 

77458 

46 

15 

57715 

81664 

59131 

80644 

60529 

79600 

61909 

78532 

63271 

77439 

45 

16 

57738 

81647 

59154 

80627 

60553 

79583 

619S2 

78514 

63293 

77421 

44 

17 

57762 

81631 

59178 

80610 

60576 

79565 

61955 

78496 

63316 

77402 

43 

18 

57786 

81614 

59201 

80593 

60599 

79547 

61978 

78478 

63338 

77384 

42* 

19 

57'810 

81597 

59225 

80576 

60622 

79530 

62001 

78460 

63361 

77366 

41 

20 

57833 

81580 

59248 

80558 

,.60645 

79512 

62024 

78442 

63383 

77347 

40 

21 

57857 

81563 

59272 

80541 

60668 

79494 

62046 

78424 

63406 

77329 

39 

22 

57881 

81546 

59295 

80524 

60691 

79477 

62069 

78405 

63428 

77310 

38 

23 

57904 

81530 

59318 

80507 

60714 

79459 

62092 

78387 

63451 

77292 

37 

24 

57928 

81513 

59342 

80489 

60738 

79441 

62115 

78369 

63473 

77273 

36 

25 

57952 

81496 

59365 

80472 

60761 

79424 

62138 

78351 

63496 

77255 

35 

26 

57976 

81479 

59389 

80455 

60784 

79406 

62160 

78333 

63518 

77236 

34 

27 

57999 

81462 

59412 

80438 

60807 

79388 

62183 

78315 

63540 

77218 

33 

28 

58023 

81445 

59436 

80420 

60830 

79371 

62206 

78297 

63563 

77199 

32 

29 

68047 

81428 

59459 

80403 

60853 

79353 

62229 

78279 

63585 

77181 

31 

30 

58070 

81412 

59482 

80386 

60876 

79335 

62251 

T8261 

63608 

77162 

30 

31 

58094 

81395 

59506 

80368 

60899 

79318 

62274 

r8243 

63630 

77144 

29 

32 

58118 

81378 

59529 

80351 

60922 

79300 

62297 

78225 

63653 

77125 

28 

33 

58141 

81361 

59552 

80334 

60945 

79282 

62320 

78206 

63675 

77107 

27 

34 

58165 

81344 

59576 

80316 

60968 

79264 

62342 

78188 

63698 

77088 

20 

35 

58189 

81327 

59599 

80299 

60991 

79247 

62365 

78170 

63720 

77070 

25 

36 

58212 

81310 

59622 

80282 

61015 

79229 

62388 

78152 

63742 

77051 

24 

37 

58236 

81293 

59646 

80264 

61038 

79211 

62411 

78134 

63765 

77033 

23 

38 

58260 

81276 

59669 

80247 

61061 

79193 

62433 

78116 

63787 

77014 

22 

39 

58283 

81259 

59693 

80230 

61084 

79176 

62456 

78098 

63810 

76996 

21 

40 

58307 

81242 

59716 

80212 

61107 

79158 

62479 

78079 

63832 

76977 

20 

41 

58330 

81225 

59739 

80195 

61130 

79140 

62502 

78061 

63854 

76959 

19 

42 

58354 

81208 

59763 

80178 

61153 

79122 

62524 

78043 

63877 

76940 

18 

43 

58378 

81191 

59786 

80160 

61176 

79105 

62547 

78025 

63899 

76921 

17 

44 

58401 

81174 

59809 

80143 

61199 

79C87 

62570 

78007 

63922 

76903 

16 

45 

58425 

81157 

59832 

80125 

61222 

79069 

62592 

77988 

63944 

76884 

15 

46 

58449 

81140 

59856 

80108 

61245 

79051 

62615 

77970 

63966 

76866 

14 

47 

58472 

81123 

59879 

80091 

61268 

79033 

62638 

77952 

63989 

76847 

13 

48 

58496 

81106 

59902 

80073 

61291 

79016 

62660 

77934 

64011 

76828 

12 

49 

58519 

81089 

59926 

80056 

61314 

78998 

62683 

77916 

64033 

76810 

11 

•50 

58543 

81072 

59949 

80038 

61337 

78980 

62706 

77897 

64056 

76791 

10 

51 

58567 

81055 

59972 

80021 

C1360 

78962 

62728 

77879 

64078 

76772 

9 

52 

58590 

81038 

59995 

800Q3 

61383 

78944 

62751 

77861 

64100 

76754 

8 

53 

58614 

81021 

60019 

79986 

61406 

78926 

62774 

77843 

64123 

76735 

7 

54 

58637 

81004 

60042 

79968 

61429 

78908 

62796 

77824 

64145 

70717 

6 

55 

58661 

80987 

60065 

79951 

61451 

78891 

62819 

77806 

64167 

76698 

5 

56 

58684 

80970 

60089 

79934 

61474 

78873 

62842 

77788 

64190 

76679 

4 

57 

58708 

80953 

60112 

79916 

61497 

78855 

62864 

77769 

64212 

76661 

3 

58 

58731 

80936 

60135 

79899 

61520 

78837 

62887 

77751 

64234 

76642 

2 

59 

58755 

80919 

6C158 

79881 

61543 

78819 

62909 

77733 

64256 

76623 

1 

60 

58779 

80902 

60182 

79864 

61566 

78801 

62932 

77715 

64279 

76604 

0 

Cosin 

Sine" 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

f 

54 

53° 

52° 

51° 

50° 

220 


TABLE  X.— SINES  AND  COSINES. 


40° 

41 

o 

42° 

43 

e 

440 

Sine 

Cosin 

Sine  Cosin 

Sine 

Cosin 

Sine  Cosin 

Sine  Cosin 

~o 

64279 

76604 

65606 

75471 

6G913 

74314 

68200 

73135 

69466 

71934 

60 

1 

64301 

76586 

65628 

75452 

6C935 

74295 

68221 

73116 

69487 

71914 

59 

2 

64323 

7G567 

65650 

75433 

66956 

74276 

68242 

73096 

69508 

71894 

58 

3 

64346 

76548 

65672 

75414 

66978 

74256 

68264 

73076 

69529! 

71873 

57 

4 

64368 

76530 

65694 

75395 

CG999 

74237 

68285 

73056 

69549 

71853 

56 

5 

64390 

76511 

65716 

75375 

C7021 

74217 

68306 

73036 

69570 

71833 

55 

6 

64412 

76492 

65738 

75356 

67043 

74198 

68327 

73016 

69591 

71813 

54 

7 

64435 

76473 

65759 

75337 

67064 

74178 

68349 

72996 

69612 

71792 

53 

8 

61457 

7G455 

65781 

75318 

C708G 

74159 

68370 

72976 

69633 

71772 

52 

9 

64479 

76436 

65803 

75239 

67107 

74139 

68391 

72957 

69654 

71752 

51 

10 

64501 

76417 

65825 

75280 

67129 

74120 

68412 

72937 

69675 

71732 

50 

11 

64524 

76398 

65847 

75261 

67151 

74100 

68434 

72917 

69696 

71711 

49 

12 

64546 

76330 

658G9 

75241 

67172 

74080 

68455 

72897 

69717 

71G91 

43 

13 

64568 

76361 

65391 

752.22 

67194 

74061 

68476! 

72877 

69737 

71671 

47 

14 

64590 

76342 

65913 

75203 

67215 

74041 

68497 

72857 

69758 

71650 

46 

15 

64612 

76323 

65935 

75184 

67237 

74022 

68518 

72837 

69779 

71G30 

45 

16 

64635 

76304 

65956 

75165 

67258 

74002 

68539 

72817 

69800 

71610 

44 

17 

64657 

76286 

65978 

75146 

67280 

73983 

68561 

72797 

69821 

71590 

43 

18 

64679 

76267 

66000 

75126 

67301 

739G3 

68582 

72777 

69842 

71569 

42 

19 

64701 

76248 

6G022 

75107 

67323 

73944 

68603 

72757 

69862 

71549 

41 

20 

64723 

76229 

66044 

75088 

67344 

73924 

68624 

72737 

69883 

71529 

40 

21 

64746 

76210 

66066 

75069 

67366 

73904 

68645 

72717 

699041 

71508 

39 

22 

64768 

76192 

6GOS8 

75050 

67387 

738S5 

68GG6 

72G97 

69925 

71488 

38 

23 

64790 

76173 

66109 

75030 

67409 

738G5 

68G88 

72677 

69946 

71468 

37 

24 

64812 

76154 

66131 

75011 

67430 

73846 

68709 

72G57 

69966 

71447 

36 

25 

64834 

76135 

66153 

74392 

67452 

73820 

68730 

72G37 

69987 

71427 

35 

26 

64856 

76116 

66175 

74973 

67473 

73806 

68751 

72617 

70008 

71407 

34 

27 

64878 

76097 

66197 

74953 

67495 

73787 

68772 

72597 

70029 

71386 

33 

28 

64901 

76078 

66218 

74931 

67516 

7376? 

68793 

72577 

70049 

713G6 

32 

29 

64923 

76059 

66240 

74915 

67538 

73747 

68814 

72557 

70070 

71345 

31 

30 

64945 

76041 

66262 

74896 

67559 

73728 

68835 

72537 

70091 

ri325 

30 

31 

64967 

76022 

66284 

74876 

67580 

73708 

68857 

72517 

70112 

71305 

29 

32 

64989 

76003 

66306 

74857 

67602 

73688 

68878 

72497 

70132 

71284 

28 

33 

65011 

75984 

66327 

74833 

67623 

73669 

68899 

72477 

70153 

71264 

27 

34 

65033 

75965 

66349 

74818 

67645 

73649 

68920 

72457 

70174 

71243 

26 

35 

65055 

75946 

66371 

74709 

67666 

73629 

68941 

72437 

70195 

71223 

35 

36 

65077 

75927 

66393 

74780 

67688 

73610 

68962 

72417 

7C215 

71203 

24 

37 

65100 

75908 

66414 

74700 

67709 

73590 

68983 

72397 

70236 

71182 

23 

38 

65122 

75889 

66436 

74741 

67730 

73570 

69004 

72377 

70257 

71162 

22 

39 

65144 

75870 

6G45S 

74722 

67752 

73551 

69025 

72357 

70277 

71141 

21 

40 

65166 

75851 

66480 

74703 

67773 

73531 

69046 

72337 

70298 

71121 

20 

41 

65188 

75832 

66501 

74683 

67795 

73511 

69067 

72317 

70319 

71100 

19 

42 

'65210 

75813 

66523 

74664 

67816 

73491 

69088 

72297 

70339 

71080 

18 

43 

65232 

75794 

6G545 

74844 

67837 

73475 

69109 

70077 

70360 

71059 

17 

44 

65254 

75775 

66566 

74625 

67859 

73452 

69130 

72257 

70381 

71039 

16 

45 

65276 

75756 

66588 

74606 

67880 

73432 

69151 

72236 

70401 

71019 

15 

46 

65298 

75738 

66610 

74586 

67901 

73413 

69172 

72216 

70422 

70998 

14 

47 

65320 

75719 

66632 

74567 

67923 

73393 

69193 

72196 

70443 

70978 

13 

48 

G5342 

75700 

66653 

74548 

67944 

73373 

69214 

72176 

70463 

70957 

12 

49 

G5364 

75680 

66675 

74528 

67965 

73353 

69235 

72156 

70484 

70937 

11 

50 

65386 

75661 

66697 

74509 

67987 

73333 

69256 

72136 

70505 

70916 

10 

51 

65408 

75642 

66718 

74489 

68008 

73314 

69277 

72116 

70525 

70896 

9 

52 

65430 

75623 

66740 

74470 

68029 

73294 

69298 

72095 

70546 

70875!  8 

53 

65452 

75604 

66762 

74451 

68051 

73274 

69319 

72075 

70567 

70855  i  7 

54 

65474 

75585 

66783 

74431 

68072 

73254 

69340 

72055 

70587 

70834  6 

55 

65496 

75566 

66805 

74412 

68093 

73234 

69361 

72035 

70608 

70813|  5 

56 

65518 

75547 

66827 

74392 

68115 

73215 

69382 

72015 

70628 

707931  4 

57 

65540 

75528 

66848 

74373 

68136 

73195 

69403 

71995 

70649 

70772 

3 

58 

65562 

75509 

66870 

74353 

68157 

73175 

69424 

71974 

70670 

70752 

2 

59 

65584 

75490 

66891 

74334 

68179 

73155 

69445 

71954 

70690 

70731 

1 

60 

65606 

75471 

66913 

74314 

68200 

73135 

69466 

71934 

70711 

70711 

0 

/ 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

Cosin 

Sine 

/ 

49° 

48° 

47° 

46° 

45 

0 

221 


NATURAL    SECANTS    AND    CO-SECANTS. 


Natural  Secants  and  Co  -secants. 

o°     it    1° 

2° 

3° 

SECANT. 

CO-SECANT.  1 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-BEC'T. 

SECANT. 

CO-SEC'T. 

Infinite. 

I.OOOI 

57-299 

i.  0006 

28.654 

1.0014 

19.  107 

3437-7 

.0001 

6-359 

.0006 

8.417 

.0014 

9.002 

1718.9 

.0002 

5-45 

.0006 

8.184 

.0014 

8.897 

145-9 

.0002 

4-57 

.0006 

7-955 

.0014 

8-794 

859.44 

.0002 

3-7i8 

.0006 

7-73 

.0014 

8.692 

687.55 

1.0002 

52.891 

1.0007 

27.508 

1.0014 

18.591 

572.96 

.OOO2 

2.09 

.0007 

7.29 

.0015 

8.491 

491.11 

•  OOO2 

1-313 

.0007 

7-075 

.0015 

8-393 

29.72 

.0002 

0-558 

.0007 

6.864 

.0015 

8.295 

381.97 

.OOO2 

49.826 

.0007 

6.655 

.0015 

8.198 

343-77 

1.0002 

49.114 

1.0007 

26.45 

1.0015 

18.103 

12.52 

.OOO2 

8.422 

.0007 

6.249 

.0015 

8.008 

286.48 

.0002 

7-75 

.0007 

6.05 

.0016 

7.914 

64.44 

.OOO2 

7.096 

.0007 

5-854 

.0016 

7.821 

45-55 

.0002 

6.46 

.0008 

5-66i 

.0016 

7-73 

229.18 

I.OOO2 

45.84 

1.0008 

25-471 

i.  0016 

17-639 

14-86 

.0002 

5-237 

.0008 

5-284 

.0016 

7-549 

02.22 

.OOO2 

4-65 

.0008 

5-i 

.0016 

7-46 

190.99 

.0002 

4-077 

.0008 

4.918 

.0017 

7-372 

80.73 

.OOO3 

3-52 

.0008 

4-739 

.0017 

7.285 

171.89 

1.0003 

42.976 

1.0008 

24.562 

1.0017 

17.198 

63-7 

.0003 

2-445 

.0008 

4-358 

.0017 

7-"3 

56.26 

.0003 

1.928 

.0008 

4.216 

.0017 

7.028 

49-47 

.0003 

1.423 

.0009 

4.047 

.0017 

6-944 

43-24 

.0003 

40-93 

.0009 

3.88 

.0018 

6.861 

i37-5i 

I.OOO3 

40.448 

1.0009 

23.716 

1.0018 

16.779 

32.22 

.0003 

39.978 

.0009 

3-553 

.0018 

6.698 

27.32 

.0003 

9.518 

.0009 

3-393 

.0018 

6.617 

22.78 

.0003 

9.069 

.0009 

3-235 

.0018 

6.538 

18.54 

.0003 

8.631 

.0009 

3-079 

.0018 

6-459 

"4-59 

1.0003 

38.201 

1.0009 

22.925 

1.0019 

16.38 

10.9 

.0003 

7.782 

.001 

2.774 

.0019 

6.303 

07-43 

.0003 

7-371 

.001 

2.624 

.0019 

6.226 

04.17 

.0004 

6.969 

.001 

2-476 

.0019 

6.15 

01.  II 

.0004 

6.576 

.001 

2-33 

.0019 

6.075 

98.223 

I.OOO4 

36.191 

1.  001 

22.186 

1.0019 

16 

5-495 

.0004 

5.814 

.001 

2.044 

.002 

5-926 

2.914 

.0004 

5-445 

.001 

1.904 

.002 

5-853 

.0001 

2.469 

.0004 

5-084 

.001 

1-765 

,OO2 

5-78 

.0001 

88.  149 

.0004 

4.729 

.0011 

1.629 

.002 

5.708 

I.OOOI 

85.946 

1.0004 

34-382 

I.OOII 

21.494 

I.OO2 

J5-637 

.0001 

3-849 

.0004 

4.042 

.0011 

1.36 

.0021 

5-566 

.0001 

1-853 

.0004 

3.708 

.OOII 

1.228 

.OO2I 

5-496 

.0001 

79-95 

.0004 

3-38i 

.0011 

1.098 

.0021 

5-427 

.0001 

8-133 

.0004 

3.06 

.OOII 

20.97 

.OO2I 

5-358 

I.OOOI 

76.396 

1.0005 

32-745 

I.OOII 

20.843 

1.  0021 

15-29 

.0001 

4.736 

.0005 

2-437 

.0012 

0.717 

.OO22 

5-222 

.0001 

3.146 

.0005 

2.134 

.0012 

0-593 

.0022 

5-155 

.0001 

1.622 

.0005 

1-836 

.0012 

0.471 

.OO22 

5-089 

.0001 

1.16 

.0005 

1-544 

.0012 

0-35 

.0022 

5.023 

I.OOOI 

68-757 

I.OOO5 

3I-257 

I.OOI2 

20.23 

I.OO22 

14.958 

.0001 

7.409 

.0005 

30.976 

.0012 

O.  112 

.0023 

4-893 

.0001 

6.113 

.0005 

0.699 

.0012 

19-995 

.0023 

4.829 

.0001 

4.866 

.0005 

0.428 

.0013 

9.88 

.OO23 

4-765 

.0001 

3-664 

.0005 

o.  161 

•0013 

9.766 

.0023 

4.702 

I.OOOI 

62.507 

1.0005 

29.899 

1.0013 

I9-653 

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Co-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T.  SECANT. 

CO-SEC'T. 

SECANT. 

89° 

88° 

87° 

860 

From  Haswell's  "Engineering."              Copyright,  1884,  by  Harper  &  Brothers 

222 

NATURAL    SECANTS   AND   CO-SECANTS. 


40 

50 

60 

70 

SECANT. 

CO-SKC'T.  i 

SECANT. 

CO-SKC'T. 

SECANT. 

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SKCANT. 

CO-SKC'T. 

1.0024 

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CO-SKC'T.;  SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T 

SKCANT. 

850 

840 

83° 

820 

From  Haswell's  "  Engineering."               Copyright,  1884,  hy  Harper  &  Brothers 

223 

NATURAL   SECANTS   AND   CO-SECANTS. 


8° 

90 

10°      11     11° 

t 

SECANT. 

CO-SEC'T 

SECANT. 

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SECANT. 

CO-SEC'T. 

SECANT. 

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CO-SKC'T. 

SECANT. 

CO-SEC'T.  SECANT. 

CO-SBC'T. 

SECANT. 

CO-SKC'T.  :  SECANT. 

|     81° 

80° 

790 

78° 

From  Haswell's  "  Engineering."              Copyright,  1884,  by  Harper  &  Brothers. 

224 

NATURAL    SECANTS    AND    CO-SECANTS. 


120 

13° 

140 

150 

SECANT. 

CO-S«C'T. 

SECANT. 

CO-SEC'T. 

SlC  A  NT. 

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1.026 

4-4736 

1.0302 

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1.0349 

3.8848 

1.0399 

3.6464 

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1.0263 

4-4454 

1.0306 

4-I336 

1-0353 

3-8637 

1.0403 

3.6279 

CO-SEC'T. 

SECANT. 

CO-SEC'T.  SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

770 

760 

75° 

74° 

From  Bagwell's  "Engineering."               Copyright,  18S4J  by  Harper  *  Brother 

225 


NATURAL    SECANTS    AND    CO-SECANTS. 


16° 

170 

18° 

19° 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T 

SECANT.  |  CO-SEC'T. 

SECANT. 

CO-SEC'T. 

1.0403 

3.6279 

1-0457 

3-4203 

1-0515 

3.2361 

1.0576 

3-°7I5 

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CO-SEC'T.  SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T.  SECANT,  j  CO-SEC'T. 

SECANT. 

730 

720     | 

71°     Jl     70° 

bruin  Haswell's  "  .Engineering."               Copyright,  1884,  by  Harper  &  Brothers. 

226 

NATURAL   SECANTS   AND   CO-SECANTS. 


20° 

SHCANT.  |  CO-SEC  'T 

2 

SBC  ANT. 

L° 

CO-SKC'T 

2 

SECANT. 

2° 
CO-SEC'T. 

2 

SECANT. 

3° 
CO-SKC'T. 

1.0642 

2.9238 

1.0711 

2.7904 

1.0785 

2.6695 

1.0864 

2-5593 

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CO-SEC'T. 

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CO-SEC'T. 

SBCANT. 

CO-SEC'T.  !  SECANT. 

JO-SKC'T. 

SECANT. 

69° 

68° 

67° 

66° 

rom  Haswell's  ••  Engineering."               Copyright,  1884,  by  Harper  &  Brother*. 

227 

NATURAL    SECANTS   AND    CO-SECANTS. 


24° 

25° 

26°       I!      270 

SECANT. 

CO-SEC'T 

SECANT. 

CO-SEC'T 

SECANT.  CO-SEC'T.  1  1  SECANT.  |  CO-SKC'T. 

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2-1359 

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•3721 

.112 

.2866 

.!2I7 

.2077 

•I3I9 

•1347 

.1029 

.3706 

.1121 

•2853 

.I2l8 

.2065 

.132 

•!335 

.1031 

.3691 

.1123 

.2839 

.122 

.2052 

.1322 

.1324 

.1032 

.1124 

.2825 

.1222 

•2039 

.1324 

.1312 

I.I034 

2.3662 

I.II26 

2.2812 

I.I223 

2.2027 

1.1326 

2.13 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

65° 

64° 

63°     | 

620 

From  Haswell's  "  Engiueering."               Copyright,  1=>84,  by  Harper  &  Brothers. 

228 

NATURAL    SECANTS    AND    CO-SECANTS. 


28° 

290 

30° 

310 

SECANT. 

CO-SKC'T. 

SECANT. 

CO-SBC'T. 

SECANT. 

CO-SKC'T. 

SECANT. 

CO-SKC'T. 

1.1326 

2-13 

I-I433 

2.0627 

I-I547 

2 

1.  1666 

1.9416 

•1327 

.1289 

•1435 

.0616 

•1549 

1.999 

.1668 

9407 

.1329 

.1277 

•1437 

.0605 

•1551 

.998 

.167 

•9397 

•1331 

.1266 

•1439 

•0594 

•»553 

•997 

.1672 

-9388 

•!333 

•1254 

.1441 

•0583 

•  1555 

.996 

.1674 

•9378 

i-i334 

2.  1242 

I-I443 

2-0573 

i-i557 

1-995 

i  1676 

i  9369 

.1336 

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•1445 

.0562 

•1559 

•994 

.1678 

•936 

•1338 

.1219 

.1446 

•0551 

.1561 

•993 

.1681 

•935 

•134 

.1208 

.1448 

•054 

.1562 

.992 

.1683 

•9341 

•I341 

.  1196 

•145 

•053 

.1564 

.991 

.1685 

•9332 

I-I343 

2.1185 

1.1452 

2.0519 

1.1566 

1.99 

1.1687 

1.9322 

'1345 

-"73 

•1454 

.0508 

.1568 

.989 

.1689 

•9313 

•I347 

.1162 

.1456 

.0498 

•!57 

.988 

1691 

•9304 

•!349 

•"5 

.1458 

.0487 

•1572 

.987 

.1693 

•9295 

•J35 

•"39 

•1459 

.0476 

•1574 

.986 

.1695 

.9285 

1-1352 

2.1127 

1.1461 

2.0466 

1.1576 

1.985 

1.1697 

1.9276 

•1354 

.1116 

.1463 

•0455 

•1578 

.984 

.1699 

.9267 

•i356 

.1104 

.1465 

.0444 

.158 

•983 

.1701 

.9258 

•1357 

.1093 

.1467 

•°434 

.1582 

.982 

•!703 

.9248 

•1359 

.1082 

.1469 

•  0423 

•1584 

.9811 

.1705 

•9239 

1.1361 

2.107 

1.1471 

2.0413 

1.1586 

1.9801 

1.1707 

1.923 

•1363 

.1059 

•1473 

.0402 

.1588 

.9791 

.'1709 

.9221 

•1365 

.1048 

.1474 

.0392 

•159 

.9781 

.1712 

.9212 

.1366 

.1036 

.1476 

.0381 

.1592 

.9771 

•i7J4 

.9203 

.1368 

1025 

.1478 

•°37 

•1594 

.9761 

.1716 

•9'93 

i-i37 

2.1014 

1.148 

2.036 

1.  1596 

1-9752 

1.1718 

1.9184 

•1372 

.1002 

.1482 

•°349 

.1598 

.9742 

.172 

•9'75 

•J373 

.0991 

.1484 

•0339 

.16 

•9732 

.1722 

.9166 

•1375 
•*377 

.098 
.0969 

.1486 
.1488 

.0329 
.0318 

.1602 
.  1604 

.9722 
•97i3 

.1724 
.1726 

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.9148 

I-I379 

2.0957 

1.1489 

2.0308 

1.  1606 

I-9703 

1.1728 

i-9'39 

.1381 

.0946 

.1491 

•0297 

.1608 

.9693 

•173 

•9*3 

.1382 

•0935 

•I493 

.0287 

.161 

.9683 

•1732 

.9121 

.1384 

.0924 

•*495 

.0276 

.1612 

.9674 

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1.1388 

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2.0901 

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1.1499 

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2.0256 

.1614 
1.  1616 

.9664 
1.9654 

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i-i739 

.9102 
1.9093 

•139 

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.1501 

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-9625 

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.0857 

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.0214. 

.  1624 

.9616 

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•9°57 

I-.I397 

2.0846 

1.1508 

2.0204 

1.1626 

1.9606 

1.1749 

1.9048 

•1399 

•0835 

•151 

•  0194 

.1628 

•9596 

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9°39 

.1401 

.0824 

.1512 

•  0183 

.163 

•9587 

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.  1402 

.0812 

•1514 

•0173 

.1632 

•9577 

•1756 

.9021 

.1404 

.0801 

•  1516 

.0163 

.1634 

.9568 

•1758 

.9013 

1.  1406 

2.079 

1.1518 

2.0152 

1.1636 

1-9558 

1.176 

.1408 

.0779 

.152 

.0142 

•  1638 

•9549 

.1762 

.8995 

.141 

.0768 

.1522 

.0132 

.164 

•9539 

.1764 

.8986 

.1411 

•  0757 

.1524 

.0122 

.1642 

•953 

.1766 

.8977 

•HI3 
1.1415 

.0746 
2-0735 

.1526 
1-1528 

.OIII 
2.OIOI 

•l6>4 
1.  1646 

•952 
i-95i 

.1768 
1.177 

.8968 
1.8959 

.1417 

.0725 

•153 

.0091 

.1648 

.9501 

.1772 

•895 

.1419 

.0714 

•I531 

.Oo8l 

.165 

.9491 

•1775 

.8941 

.1421 

.0703 

•1533 

.0071 

.1652 

.9482 

.1777 

•  8932 

.1422 

.0692 

•1535 

.Oo6l 

.1654 

•9473 

.1779 

.8924 

1.1424 

2.0681 

i-i537 

2.005 

1.1656 

1.9463 

1.1781 

1-8915 

.1426 

.067 

•1539 

.004 

.1658 

•9454 

•1783 

.8906 

.1428 

.0659 

•i54i 

.003 

.166 

•9444 

•1785 

.8897 

•143 

.0648 

•1543 

.OO2 

.1662 

•9435 

.1787 

.8888 

.1432 

.0637 

•1545 

.OOI 

.1664 

•9425 

.179 

.8879 

J-I433 

2.0627 

i-i547 

2 

1.  1666 

1.9416 

1.1792 

1.8871 

CO-SEC'T 

SECANT. 

CO-SKC'T 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

61° 

60° 

59° 

58° 

From  Haswell's  "  Engineering."               Copyright,  1884,  by  Harper  &  Brother* 

229 

NATURAL    SECANTS    AND    CO-SECANTS. 


31 

jo 

3: 

$o 

3' 

1° 

3d 

o 

SECANT. 

CO-SEC'T. 

SKCANT. 

CO-SKC'T. 

SECANT. 

CO-SKC'T. 

SKCANT. 

CO-SKC'T. 

1.1792 

1.8871 

1.1924 

1.8361 

1.2062 

1.7883 

1.2208 

*-7434 

.1794 

.8862 

.  1926 

.8352 

.2064 

•7875 

.221 

.7427 

.1796 

•8853 

.1928 

•8344 

.2067 

.7867 

.2213 

•742 

.1798 

.8844 

•193 

8336 

.2069 

.786 

.2215 

•74i3 

.18 

.8836 

•1933 

.8328 

.2072 

.7852 

.2218 

•7405 

1.  1802 

1.8827 

I-I935 

1.832 

1.2074 

1.7844 

1.222 

I-7398 

-1805 

.8818 

•1937 

.8311 

2076 

7837 

.  223 

•7391 

.1807 

.8809 

•*939 

•8303 

.2079 

.7829 

.  225 

•7384 

.1809 

.8801 

.1942 

.8295 

.2081 

.7821 

.  228 

•7377 

.1811 

.8792 

.1944 

.8287 

.2083 

.7814 

23 

•7369 

1.1813 

1.8783 

1.1946 

1.8279 

1.2086 

1.7806 

I-  233 

1.7362 

.1815 

.8785 

.1948 

.8271 

.2088 

.7798 

•  235 

•7355 

.1818 

.8766 

•I951 

8263 

.2091 

.7791 

•  238 

•7348 

.182 
.1822 

•8757 
.8749 

•1953 
•1955 

•8255 
.8246 

•2093 
•  2095 

•7783 
.7776 

.  24 
•  243 

•7341 
•7334 

1.1824 

1.874 

1.1958 

1.8238 

1.2098 

1.7768 

I.  245 

I-7327 

.1826 

•8731 

.196 

.823 

'21 

.776 

.  248 

.1828 

.8723 

.  1962 

.8222 

.2103 

•7753 

•  25 

•7312 

.1831 

.8714 

.1964 

.8214 

•2105 

•7745 

•  253 

•73°5 

•1833 

.8706 

.1967 

•  8206 

.2107 

•7738 

•  255 

.7298 

1-1835 

1.8697 

1.1969 

1.8198 

1.  211 

J-773 

I-  258 

1.7291 

•1837 

.8688 

.1971 

.819 

.2112 

•7723 

.226 

.7284 

.1839 

.868 

.1974 

.8182 

.2115 

•77J5 

.2263 

•7277 

.1841 

.8671 

.1976 

.8174 

.2II7 

.7708 

•2265 

•727 

.1844 

.8663 

.1978 

.8166 

.2119 

•77 

.2268 

.7263 

1.1846 

1.8654 

1.198 

1.8158 

I.  2122 

1-7693 

1.227 

1.7256 

.1848 

.8646 

.1983 

-815 

.2124 

.7685 

.2273 

•7249 

.185 

-8637 

.1985 

.8142 

.2127 

.7678 

.2276 

.7242 

.1852 

.8629 

.1987 

-8i34 

.2129 

•767 

.2278 

•7234 

•1855 

.862 

.199 

.8126 

.2132 

.7663 

.228l 

•7227 

1.1857 

1.8611 

1.1992 

1.8118 

I-2I34 

1-7655 

1.2283 

1.722 

.1859 

.8603 

.1994 

.811 

.2136 

.7648 

.2286 

•7213 

.1861 

•8595 

.1997 

.8102 

.2139 

.764 

.2288 

.7206 

.1863 

.8586 

.1999 

.8094 

.2141 

•7633 

.2291 

.7199 

.1866 

.8578 

.2001 

.8086 

.2144 

.7625 

.2293 

.7192 

1.  1868 

1.8569 

1  .  2004 

1.8078 

1.2146 

1.7618 

1.2296 

1.7185 

.187 

•  8561 

.2OO6 

.807 

.2149 

.761 

.2298 

.7178 

.1872 

-8552 

.2008 

.8062 

.2151 

.7603 

.23OI 

.7171 

.1874 

•8544 

.201 

•  8054 

•2153 

.2304 

.7164 

.1877 

•8535 

.2013 

.8047 

•2156 

.7588 

.2306 

•7*57 

1.1879 

1.8527 

I.2OI5 

1.8039 

1.2158 

1.7581 

1.2309 

.1881 

.8519 

.2017 

.8031 

.2l6l 

•7573 

.2311 

.7144 

.1883 

.851 

.202 

.8023 

.2163 

.7566 

.2314 

•7137 

.1886 

.8502 

.2022 

.8015 

.2166 

•7559 

.2316 

.1888 

•8493 

.2024 

.8007 

.2168 

•7551 

.2319 

.7123 

1.189 

1.8485 

I.  2027 

1.7999 

I.2I7I 

J-7544 

1.2322 

1.7116 

.1892 

.8477 

.2029 

.7992 

•2173 

•7537 

.2324 

.7109 

.1894 

.8468 

.2031 

.7984 

•2175 

•7529 

.2327 

.7102 

.1897 

.846 

.2034 

.7976 

.2178 

•7522 

.2329 

•7095 

.1899 

•  8452 

.2036 

.7968 

.218 

•7514 

•2332 

.7088 

1.  1901 

1.8443 

1.2039 

1.796 

1.2183 

I-7507 

1-2335 

1.7081 

.1903 

•8435 

.2041 

•7953 

.2185 

•75 

•2337 

•7°75 

.1906 

.8427 

.2043 

•7945 

.2188 

•7493 

•234 

.7068 

.1908 

.8418 

.2046 

•7937 

.219 

7485 

.2342 

.7061 

.191 

.841 

.2048 

.7929 

.2193 

.7478 

•2345 

•7054 

1.1912 

1.8402 

1.205 

1.7921 

1.2195 

1.7471 

1.2348 

1.7047 

•19'S 

•8394 

•2053 

.7914 

.2198 

•7463 

•235 

.704 

.1917 

.8385 

•2055 

.7906 

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•7456 

•2353 

•7033 

.1919 

•8377 

.2057 

.7898 

.2203 

•7449 

•2355 

.7027 

.1921 

-8369 

.206 

.7891 

.2205 

.7442 

•2358 

.702 

1.1922 

1.8361 

1.2062 

1.7883 

1.  2208 

J-7434 

1.2361 

1.7013 

CO-SEC'T. 

SECANT. 

CO-SKC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SKC'T. 

SECANT. 

5' 

r« 

5C 

o 

5v 

r»«  

)° 

64 

L° 

i'rojn  Haswell's  "  Engineering." 


Copyright,  1884,  by  Harper  <5t  Brothers. 


230 


NATURAL    SECANTS    AND    CO-SECANTS. 


36° 

370 

38° 

39° 

SECANT. 

CO-SEC'T. 

SKCANT.  CO-SEC'T. 

SECANT.  |  CO-SEC'T. 

SECANT.  CO-SEC'T. 

1.2361 

1.7013 

1.2521 

1.6616 

1.269 

1.6243 

1.2867 

1.589 

•2363 

.7006 

.2524 

.661 

.2693 

•6237 

.2871 

-5884 

.2366 

•6999 

.2527 

.6603 

.2696 

•6231 

.2874 

•5879 

.2368 

•6993 

•253 

•6597 

.2699 

.6224 

.2877 

•5873 

•2371 

.6986 

.6591 

.2702 

.6218 

.288 

.5867 

1-2374 

1.6979 

1-2535 

1.6584 

1.2705 

1.6212 

1.2883 

1.5862 

.2376 

.6972 

.2538 

.6578 

.2707 

.6206 

.2886 

•5856 

•2379 

.6965 

.2541 

•6572 

.271 

.62 

.2889 

•585 

.2382 

•6959 

•2543 

•6565 

•2713 

.6194 

.2892 

•5845 

.2384 

.6952 

.2546 

•6559 

.2716 

.6188 

.2895 

•5839 

1.2387 

1.6945 

1-2549 

1-6552 

1.2719 

1.6182 

1.2898 

1-5833 

.2389 

.6938 

•2552 

.6546 

.2722 

.6176 

.2901 

.5828 

•  2392 

.6932 

•2554 

•654 

.2725 

.617 

.2904 

.5822 

•2395 

•  6925 

•2557 

•6533 

.2728 

.6164 

.2907 

.5816 

•2397 

.6918 

.256 

.6527 

•2731 

.6159 

.291 

.5811 

1.24 

1.6912 

1.2563 

1.6521 

1-2734 

1.6153 

1.2913   1.5805 

.2403 

.6905 

•2565 

.6514 

•2737 

.6147 

.2916    .5799 

•2405 

.6898 

.2568 

.6508 

•2739 

.6141 

.2919    .5794 

.2408 

.6891 

•2571 

.6502 

.2742 

•6i35 

.29*    .5788 

.2411 

.6885 

•2574 

.6496 

•2745 

.6129 

.2926  i   .5783 

1.2413 

1.6878 

1-2577 

1.6489 

1.2748 

1.6123 

1.2929 

1-5777 

.2416 

.6871 

•2579 

.6483 

•2751 

.6117 

.2932 

.5771 

.2419 

.6865 

.2582 

•6477 

•2754 

.6111 

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.5766 

.2421 

.6858 

•2585 

.647 

•2757 

•  6105 

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.2424 

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.2588 

.6464 

.276 

.6099 

.2941 

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1.2427 

1-6845 

1.2591 

1.6458 

1.2763 

1.6093 

1.2944 

J-5749 

.2429 

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•  6452 

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•2947 

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•2953 

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•6433 

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1.244 

1.6812 

1.2605 

1.6427 

1.2778 

1.6064 

1.296 

1.5721 

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.6805 

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.642 

.2781 

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.2963 

.5716 

•2445 

.6798 

.261 

.6414 

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.6052 

.2966 

•571 

.2448 

.6792 

.2613 

'  .6408 

.2787 

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.2451 

.6785 

.2616 

.6402 

.279 

.604 

.2972 

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1-2453 

1.6779 

1.2619 

1.6396 

1-2793 

1.6034 

1-2975 

1.5694 

.2456 

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.2464 

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.263 

•6371 

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.2988 

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1.2467 

1.6746 

1.2633 

1-6365 

1.2807 

1.6005 

1.2991 

1.5666 

.247 

•6739 

.2636 

•6359 

.281 

.6 

•2994 

•  5661 

.2472 

.2639 

•6352 

.2813 

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.6726 

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.2473 

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.2644 

•634 

.2819 

.5982 

•3003 

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1.248 

1.6713 

1.2647 

I.6334 

1.2822 

I-5976 

1.3006 

1-5639 

.2483 

.6707 

.265 

.6328 

.2825 

.301 

•5633 

.2486 

.67 

•2653 

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.2828 

•5965 

3OI3 

.5628 

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•5953 

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1.2494 

i.  6681 

1.2661 

1.6303 

1.2837 

1-5947 

1.3022 

1.5611 

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•5942 

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.6668 

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•5936 

•  3029 

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.6285 

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•  3032 

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1.2508 

1.6648 

1.2676 

1.6273 

1.2852 

1-3038 

1-5584 

•  251 

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.2858 

•59°7 

•3°44 

•5573 

•2516 

.6629 

.2684 

•6255 

.2861 

.3048 

•5568 

•2519 

.6623 

.2687 

.6249 

.2864 

.5896 

•3051 

•5563 

1.2521 

i.  6616 

1.269 

1.6243 

1.2867 

1.589 

I-3°54 

I-5557 

CO-SEC'T 

SECANT. 

CO-SEC'T 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

530 

520 

510 

500 

From  Haswell's  "  Engineering."              Copyright,  1884,  by  Harper  &  Brothera 

231 

NATURAL    SECANTS    AND   CO-SECANTS. 


40° 

41° 

42° 

43° 

SECANT. 

CO-SKC'T. 

SECANT. 

CO-SHC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

CO-SEC'T. 

I-  3°54 

1-5557 

L325 

1.5242 

I-3456 

1-4945 

I-3673 

1.4663 

:p7 

•5552 
•5546 

•3253 

•5237 
•5232 

.346 
•3463 

•494 
•4935 

•3677 
•3681 

.4658 
•4654 

.3064 

•5541 

•326 

•5227 

•3467 

•493 

.3684 

•4649 

.3067 

•5536 

•3263 

.5222 

•347 

•4925 

.3688 

•4644 

1.307 

!-553 

1.3267 

1.5217 

1-3474 

1.4921 

1.3692 

1.464 

•3°73 

•5525 

•327 

•  5212 

•3477 

.4916 

•3695 

•4635 

.3076 

•552 

•3274 

.5207 

.3481 

•49" 

•3699 

.4631 

.308 

•5514 

.3277 

.5202 

•3485 

.4906 

•3703 

.4626 

•3083 

•5509 

•328 

•5197 

.3488 

.4901 

•37°7 

.4622 

1.3086 

I-5503 

1.3284 

1.5192 

1.3492 

1.4897 

I-37I 

1.4617 

•3089 

•5498 

.3287 

•5187 

•3495 

.4892 

•37M 

.4613 

•  3092 

•5493 

•329 

.5182 

•3499 

.4887 

.4608 

•  3096 

•5487 

•3294 

•5177 

•3502 

.4882 

.3722 

.4604 

•3099 

•  5482 

•3297 

•  5171 

•35o6 

.4877 

•3725 

•4599 

1.3102 

•1-5477 

i-3301 

1.5166 

1-3509 

1-4873 

I-3729 

1-4595 

•3105 

•5471 

•3304 

.5161 

•3513 

.4868 

•3733 

•459 

.3109 

.5466 

•3307 

•5156 

•3517 

.4863 

•3737 

•  4586 

.3112 

.5461 

•33" 

.5151 

•352 

•  4858 

•374 

.4581 

•3"5 

•5456 

.5146 

•3524 

•4854 

•3744 

•4577 

1.3118 

1-545 

1.3318 

1.5141 

1-3527 

1.4849 

I-3748 

I-4572 

.3121 

•5445 

•3321 

•5136 

•3531 

•4844 

•3752 

4568 

•3125 

•544 

•3324 

•3534 

•4839 

•3756 

•4563 

.3128 

•5434 

•3328 

.5126 

•3538 

•4835 

•3759 

•4559 

•3I3I 

•5429 

•3331 

•  5121 

•3542 

•483 

•3763 

•4554 

I-3I34 

1-5424 

1-3335 

1.5116 

1-3545 

1.4825 

1-3767 

1-455 

•3138 

•  5419 

•3338 

.5111 

•3549 

.4821 

•3771 

•4545 

•314I 

•54I3 

•3342 

.5106 

•3552 

.4816 

•3774 

•4541 

•3J44 

•  5408 

•3345 

.5101 

•3556 

.4811 

•3778 

•4536 

.3148 

•54°3 

•3348 

•  5096 

•356 

.4806 

•3782 

4532 

I-5398 

1-3352 

1.5092 

1-3563 

1.4802 

1-3786 

14527 

•3154 

•5392 

•3355 

.5087 

•3567 

•4797 

•379 

•4523 

•5387 

•3359 

.5082 

•3571 

.4792 

•3794 

.4518 

.3161 

•5382 

•  3362 

•5077 

•3574 

.4788 

•3797 

•45H 

.3164 

•5377 

•3366 

.5072 

•3578 

•4783 

•3801 

•451 

1.3167 

1-3369 

1.5067 

i-358i 

1.4778 

1-3805 

I-4505 

•3J7 

'•5366 

•3372 

.5062 

•3585 

•4774 

.3809 

.4501 

-536i 

•3376 

5057 

•3589 

.4769 

-3813 

•4496 

•3177 

•5356 

•3379 

•5052 

•3592 

•4764 

-3816 

.4492 

•  318 

•5351 

•3383 

•5°47 

-476 

•  382 

•4487 

1-3184 

1-5345 

1-3386 

1.5042 

1-4755 

1.3824 

1.4483 

•3187 

•534 

•339 

•5037 

•  3603 

•475 

.3828 

•4479 

•3I9 

•5335 

•3393 

.5032 

•3607 

.4746 

•3832 

•4474 

•3J93 

•533 

•3397 

.5027 

.36" 

.4741 

-3836 

•447 

•5325 

•34 

.5022 

.3614 

•4736 

•3839 

•4465 

1.32 

1.3404 

1.5018 

1.3618 

1-4732 

1-3843 

1.4461 

•3203 

•53J4 

•3407 

•5013 

.3622 

•4727 

•3847 

•4457 

.3207 

•5309 

•34" 

.5008 

•3625 

•4723 

•3851 

•4452 

.321 

•53°4 

•34H 

•5003 

•3629 

.4718 

•3855 

.4448 

•3213 

•5299 

.3418 

.4998 

•3633 

•3859 

•4443 

1.3217 

1.5294 

1.3421 

1-4993 

1-3636 

1.4709 

1-3863 

J-4439 

•  322 

.5289 

•3425 

.4988 

•364 

•47°4 

.3867 

•4435 

•3223 

•5283 

•3428 

•4983 

•3644 

•4699 

-387 

•443 

.3227 

.5278 

•3432 

•4979 

•3647 

.4695 

•3874 

.4426 

•323 

•5273 

•3435 

•4974 

•3651 

.469 

.3878 

•  44E2 

I-3233 

1.5268 

J-3439 

1.4969 

1-3655 

1.4686 

1.3882 

1.4417 

•3237 
•324 

•5263 
•5258 

•3442 
•3446 

•4964 
•4959 

•3658 

.4681 
.4676 

.3886 
•  389 

.4408 

•3243 

•5253 

•3449 

•4954 

.'3666 

.4672 

.4404 

•3247 

•5248 

•3453 

•4949 

-3669 

.3667 

.3898 

•44 

1-325 

1.5242 

I-3456 

r-4945 

1-3673 

1.4663 

1.3902 

1-4395 

CO-SEC'T. 

SKCANT. 

CO-SEC'T. 

SKCANT. 

CO-SBC'T. 

SECANT. 

CO-SEC'T. 

SECANT. 

49° 

48° 

470 

46° 

From  Haswell's  "  Engineering."              Copyright,  1884,  by  Harper  «k  Brother* 

NATURAL   SECANTS   AND    CO-SECANTS. 


44 

to 

44 

0 

44 

to 

; 

SECANT. 

CO-SEC'T. 

f 

i 

SECANT. 

CO-SEC'T. 

/ 

1 

SECANT. 

CO-SKC'T. 

' 

0 

1.3902 

1-4395 

60 

21 

1.3984" 

I-4305 

39 

41 

1.4065 

1.4221 

19 

I 

•39°5 

•4391 

59 

22 

.3988 

.4301 

38 

42 

.4069 

.4217 

18 

c 

•39°9 

•4387 

58 

23 

•3992 

•4297 

37 

43 

•4073 

.4212 

17 

3 

•39X3 

.4382 

57 

24 

.3996 

.4292 

36 

44 

.4077 

.4208 

16 

4 

•39*7 

•4378 

56 

25 

1.4 

1.4288 

35 

45 

1.4081 

1.4204 

15 

5 

1.3921 

*-4374 

55 

26 

.4004 

.4284 

34 

46 

.4085 

.42 

M 

6 

•3925 

•437 

54 

27 

.4008 

.428 

33 

47 

.4089 

.4196 

J3 

7 

3929 

•4365 

53 

28 

.4012 

•4276 

S2 

48 

•4°93 

.4192 

12 

8 

•3933 

on"37 

.4361 

A  OC7 

52 

CT 

29 

1O 

.4016 

I  J.O2 

.4271 

3i 

49 

.4097 

.4188 

II 

IO 

9 
«o 

•  oVO/ 

1-3941 

•43j/ 

1-4352 

JX 

5° 

J^ 

31 

**^M« 

.4024 

I.  42t>7 

.4263 

3O 

29 

5^ 

51 

.4105 

.4179 

9 

ii 

•3945 

•4348 

49 

32 

.4028 

•4259 

28 

52 

.4109 

•4!75 

8 

12 

13 

•3949 
•3953 

•4344 
•4339 

48 
47 

33 
34 

.4032 
.4036 

•4254 
•425 

27 
26 

53 
54 

•4H3 
.4117 

.4171 
.4167 

I 

14 

•3957 

•4335 

46 

35 

1.404 

1.4246 

25 

55 

1.4122 

1.4163 

5 

5 

1.396 

I-433I 

45 

36 

.4044 

.4242 

24 

56 

.4126 

•4*59 

4 

6 

•3964 

•4327 

44 

37 

.4048 

.4238 

23 

57 

•4*3 

•4154 

3 

7 

.3968 

.4322 

43 

38 

.4052 

•4333 

22 

58 

•4!34 

•4*5 

2 

8 

•3972 

.4318 

42 

39 

.4056 

.4229 

21 

59 

.4138 

.4146 

I 

9 

•3976 

•43H 

4i 

40 

1.406 

1.4225 

20 

60 

1.4142 

1.4142 

0 

20 

1.398 

i-43i 

40 

/ 

CO-SEC  'T. 

SECANT. 

> 

/ 

CO-SKC'T. 

SECANT. 

; 

/ 

CO-SKC'T. 

SECANT. 

/ 

4t 

>° 

4 

)° 

4 

50 

Preceding  Table  contains  Natural  Secants  and  Co-secants  for  every 
minute  of  the  Quadrant  to  Radius  i. 

If  Degrees  are  taken  at  head  of  column,  Minutes,  Secant,  and  Co-secant 
must  be  taken  from  head  also;  and  if  they  are  taken  at  foot  of  column, 
Minutes,  etc.,  must  be  taken  from  foot  also. 

ILLUSTRATION.—  1.05  is  secant  of  17°  45'  and  co-secant  of  72°  15'. 

To    Compute    Secant   or   Co-secant   of  any   Angle. 
RULE.  —  Divide  i  by  Cosine  of  angle  for  Secant,  and  by  Sine  for  Co-secant. 
EXAMPLE  i.  —  What  is  secant  of  25°  25'? 

Cosine  of  angle  =  .903  21.    Then  i  -r-  .903  21  =  1.  1072,  Secant. 
2.  —  What  is  co-secant  of  64°  35'? 

Sine  of  angle  =  .903  21.    Then  1-^.903213=1.1072,  Co-secant. 


From  Haswell's  "Engineering. 


Copyright,  1884,  by  Harper  &  Brothers. 


233 


TABLE  XII.— TANGENTS  AND  COTANGENTS. 


0* 

1 

o 

2 

0 

3 

o 

1 

/ 

Tang 

Cotang 

Tang 

Cotang 

Tang, 

Cotang 

Tang 

Cotang 

0 

00000 

Infinite. 

01746 

57.2900 

03492 

28.6363 

05241 

19.0811 

GO 

1 

00029 

3437.75 

01775 

56.3506 

03521 

28.3994 

05270 

18.9755 

59 

2 

00058 

1718.87 

01804 

55.4415 

03550 

28.1664 

05299 

18.8711 

58 

3 

00087 

1145.92 

01833 

54.5613 

03579 

27.9372 

05328 

18.7678 

57 

4 

00116 

859.436 

01862 

53.7086 

03609 

27.7117 

05357 

18.6656 

56 

5 

00145 

687.549 

01891 

52.8821 

03638 

27.4899 

05387 

18.5645 

55 

G 

00175 

572.957 

01920 

52.0807 

03667 

27.2715 

05416 

18.4645 

54 

7 

00204 

491.106 

01949 

51.3032 

03696 

27.0566 

05445 

18.3655 

53 

8 

00233 

429.718 

01978 

50.5485 

03725 

26.8-150 

05474 

18.2677 

52 

9 

00262 

381.971 

02007 

49.8157 

03754 

26.6367 

05503 

18.1708    51 

10 

00291 

343.774 

02036 

49.1039 

03783 

26.4316 

05533 

18.0750 

50 

11 

00320 

312.521 

02066 

48.4121 

03812 

26.2296 

05562 

17.9802 

49 

12 

00349 

286.478 

02095 

47.7395 

03842 

26.0307 

05591 

17.8863 

48 

13 

00373 

2G4.441 

02124 

47.0353 

03871 

25.8348 

05620 

17.7934 

47 

14 

00407 

245.552 

02153 

46.4489 

03900 

25.6418 

05649 

17.7015 

46 

15 

00436 

229.182 

02182 

45.8294 

03929 

25.4517 

05678 

17.6106 

45 

16 

004G5 

214.858 

02211 

45.2261 

03958 

25.2644 

05708 

17.5205 

44 

IT 

00495 

202.219 

02240 

44.6386 

03987 

25.0798 

05737 

17.4314 

43 

18 

00524 

190.984 

02269 

44.0661 

04016 

24.8978 

05766 

17.3432 

42 

19 

00553 

180.932 

02298 

43.5081 

04046 

24.7185 

05795 

17.2558 

41 

20 

00583 

171.885 

02328 

42.9641 

04075 

24.5418 

05824 

17.1693 

40 

21 

00611 

163.700 

02357 

42.4335 

04104 

24.3675 

05854 

17.0837 

39 

22 

00640 

156.259 

02386 

41.9158 

04133 

24.1957 

05S83 

16.9990 

38 

23 

006G9 

149.465 

02415 

41.4106 

04162 

24.0263 

05912 

16.9150 

37 

24 

00698 

143.237 

02444 

40.9174 

04191 

23.8593 

05941 

16.8319 

36 

25 

00727 

137.507 

02473 

40.4358 

04220 

23.6945 

05970 

16.7496 

35 

20 

00756 

132.219 

02502 

39.9655 

04250 

23.5321 

05999 

16.6681 

34 

27 

00785 

127.321 

02531 

39.5059 

04279 

23.3718 

06029 

16.5874 

33 

28 

00815 

122.774 

02560 

39.0568 

04308 

23.2137 

06058 

16.5075 

32 

29 

008*^4 

118.540 

02589 

38.6177 

04337 

23.0577 

06087 

16.4283 

31 

30 

00873 

114.589 

02619 

38.1885 

04366 

22.9038 

06116 

16.3499 

30 

31 

00902 

110.892 

02648 

37.7686 

04395 

22.7519 

06145 

16.2722 

29 

oo 

00931 

107.426 

02677 

37.3579 

04424 

22.6020 

06175 

16.1952 

23 

88 

00960 

101.171 

02706 

36.9560 

04454 

22.4541 

06204 

16.1190 

27 

34 

00989 

101.107 

02735 

36.5G27 

04483 

22.3081 

06233 

16.0435 

26 

35 

01018 

98.2179 

02764 

36.1776 

04512 

22.1640 

06262 

15.9687 

25 

36 

01047 

95.4895 

02793 

35.8006 

04541 

22.0217 

06291 

15.8945 

24 

37 

01076 

92.9085 

02822 

35.4313 

04570 

21.8813 

06321 

15.8211 

23 

88 

01105 

90.4633 

02351 

35.0G95 

04599 

21.7426 

06350 

15.7483 

22 

39 

01135 

83.1436 

02881 

34.7151  t 

04628 

21.6056 

06379 

15.6762 

21 

40 

011G4 

85.9398 

02910 

34.3678 

04658 

21.4704 

06408 

15.6048 

20 

41 

01193 

83.8435 

02939 

34.0273 

04687 

21.3369 

06437 

15.5340 

19 

42 

01222 

81.8470 

02963 

33.6935 

04716 

21.2049 

06467 

15.4638 

18 

43 

01251 

79.9434 

02997 

33.3662 

04745 

21.0747 

06496 

15.3943 

17 

44 

01280 

78.1263 

03026 

83.0452 

04774 

20.9460 

06525 

15.3254 

16 

45 

01309 

76.3900 

03055 

32.7303 

04803 

20.8188 

06554 

15.2571 

15 

46 

C1338 

74.7292 

03084 

32.4213 

04833 

20.6932 

06584 

15.1893 

14 

47 

01367 

73.1390 

03114 

32.1181 

04862 

20.5691 

06613 

15.1222 

13 

48 

01396 

71.6151 

03143 

31.8205 

04891 

20.4465 

06642 

15.0557 

12 

49 

01425 

70.15,33 

03172 

31.5284 

04920 

20.3253 

06671 

14.9898 

11 

50 

01455 

68.7501 

03201 

31.2416 

04949 

20.2056 

06700 

14.9244 

10 

51 

01484 

67.4019 

03230 

30.9599 

04978 

20.0872 

06730 

14.8596 

9 

52 

01513 

66.1055 

03259 

30.6833 

05907 

19.9702 

06759 

14.7954 

8 

53 

01542 

64.8580 

03288 

30.4116 

05037 

19.8546 

06788 

14.7317 

7 

54 

01571 

63.G567 

03317 

30.1446 

05066 

19.7403 

06817 

14.6685 

6 

55 

01600 

62.4992 

03346 

29.8823 

05095 

19.6273 

06847 

14.6059 

5 

56 

01629 

61.3829 

03376 

29.6245 

05124 

19.5156 

06876 

14.5438 

4 

57 

01658 

60.3058 

03405 

29.3711 

05153 

19.4051 

06905 

14.4823 

3 

58 

01687 

59.2659 

03434 

29.1220 

05182 

19.2959 

06934 

14.4212 

2 

59 

01716 

58.2612 

03463 

28.8771 

05212 

19.1879 

06963 

14.3607 

1 

60 

01746 

57.2900 

03492 

28.6363 

05241^ 

19.0811 

06993 

14.3007 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/• 

89" 

88° 

87' 

86° 

'235 


TABLE  XII.-TANGENTS  AND  COTANGENTS. 


4° 

5° 

6° 

7° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

' 

0 

.06993 

14.3007 

08749 

11.4301 

10510 

9.51436 

'     12278 

8.14435 

00 

1 

07022 

14.2411 

08778 

11.3919 

10540 

9.48781 

12308 

8.12481 

59 

2 

07051 

14.1821 

08807 

11.3540 

10569 

9.46141 

12338 

8.10530 

58 

3 

07080 

14.1235 

08837 

11.3163 

10599 

9.43515 

12367 

8.08600 

57 

4 

07110 

14.0655 

08866 

11.2789 

10628 

9.40904 

12397 

8.06674 

56 

5 

07139 

14.0079 

08895 

11.2417 

10657 

9.38307 

12426 

8.04756 

55 

6 

07168 

13.9507 

08925 

11.2048 

10687 

9.35724 

12456 

8.02848 

54 

7 

07197 

13.8940 

08954 

11.1681 

10716 

9.33155 

12485 

8.00948 

53 

8 

07227 

13.8378 

08983 

11.1316 

10746 

9.30599 

12515 

7.99058 

52 

9 

07256 

13.7821 

09013 

11.0954 

10775 

9.28058 

12544 

7.97176 

51 

10 

07285 

13.7267 

09042 

11.0594 

10805 

9.25530 

12574 

7.95302 

50 

11 

07314 

13.6719 

09071 

11.0237 

10834 

9.23016 

12603 

7.93438 

49 

12 

07344 

13.6174 

09101 

10.9882 

108C3 

9.20516 

12633 

7.91582 

48 

13 

07373 

13.5634 

09130 

10.9529 

10893 

9.18028 

12662 

7  89734 

47 

14 

07402 

13.5098 

09159 

10.9178 

10922 

9.15554 

12692 

7.87895 

40 

15 

07431 

13.4566 

09189 

10.8829 

10952 

9.13093 

12722 

7.86064 

45 

16 

07461 

13.4039 

09218 

10.8483 

10981 

9.10646 

12751 

7.84242 

44 

17 

07490 

13.3515 

09247 

10.8139 

11011 

9.08211 

12781 

7.82428 

4,3 

18 

07519 

13.2996 

09277 

10.7797 

11040 

9.05789 

12810 

7  80622 

42 

19 

07548 

13.2480 

09306 

10.7457 

11070 

9.03379 

12840 

7.78825 

41 

20 

07578 

13.1969 

09335 

10.7119 

11099 

9.00983 

12869 

7.77035 

40 

21 

07607 

13.1461 

09365 

10.6783 

11128 

8.98598 

12899 

7.75254 

39 

22 

07636 

13.0958 

09394 

10.6450 

11158 

8.96227 

12929 

7.73480 

3H 

23 

07665 

13.0458 

09423 

10.6118 

11187 

8.93867 

12958 

7.71715 

37 

24 

07695 

12.9962 

09453 

10.5789 

11217 

8.91520 

12988 

7.69957 

30 

25 

07724 

12.9469 

09482 

10.5402 

11246 

8.89185 

13017 

7.68208 

35 

20 

07753 

12.8981 

09511 

10.5136 

11276 

8.86802 

13047 

7.66466 

34 

27 

07783 

12.8496 

09541 

10.4813 

11305 

8.84551 

13076 

7.64732 

38 

ys 

07812 

12.8014 

09570 

10.4491 

11335 

8.82252 

13106 

7.63005 

32 

29 

07841 

12.7536 

09600 

10.4172 

11364 

8.79964 

13136 

7.61287 

31 

30 

07870 

12.7062 

09029 

10.3854 

11394 

8.77689 

13165 

7.59575 

30 

31 

07899 

12.6591 

09658 

10.3538 

11423 

8.75425 

13195 

7.57872 

29 

32 

07929 

12.6124 

09088 

10.3224 

11452 

8.73172 

13224 

7.56176 

•js 

33 

07958 

12.5660 

09717 

10.2913 

11482 

8.70931 

13254 

7.54487 

27 

34 

07987 

12.5199 

09746 

10.2602 

11511 

8.68701 

13284 

7.52806 

20 

35 

08017 

12.4742 

09776 

10.2294 

11541 

8.66482 

13313 

7.51132 

25 

30 

08046 

12.4288 

09805 

10.1988 

11570 

8.64275 

13343 

7.49465    24 

37 

08075 

12.3838 

09834 

10.1683 

11600 

8.62078 

13372 

7.47806 

23 

38 

08104 

12.3390 

09864 

10.1381 

11629 

8.59893 

13402 

7.46154 

.,'.> 

39 

08134 

12.2946 

09893 

10.1080 

IK    > 

8.57718 

13432 

7.44509 

21 

40 

08163 

12.2505 

09923 

10.0780 

11688 

8.55555 

13461 

7.42871 

20 

41 

08192 

12.2067 

09952 

10.0483 

11718 

P.  53402 

13491 

7.41240 

19 

42 

08221 

12.1632 

09981 

10.0187 

11747 

8.51259 

13521 

7.39616 

18 

43 

08251 

12.1201 

10011 

9.98931 

11777 

8.49128 

13550 

7.37999 

17 

44 

08280 

12.0772 

10040 

9.96007 

11806 

8.47007 

13580 

7.30389 

16 

45 

08309 

12.0346 

10069 

9.93101 

11836 

8.44896 

13609 

7.34786 

15 

40 

08339 

11.9923 

10099 

9.90211 

11865 

8.42795 

18639 

7.33190 

14 

47 

08368 

11.9504 

10128 

9.87338 

11895 

8.40705 

13669 

7.31600 

13 

48 

08397 

11.9087 

10158 

9.84482 

11924 

8.38625 

13698 

7.30018 

12 

49 

08427 

11.8673 

10187 

9.81641 

11954 

8.36555 

13728 

7.28442 

11 

50 

08456 

11.8262 

10216 

9.78817 

11983 

8.34496 

13758 

7.26873 

10 

51 

08485 

11.7853 

10246 

8.76009 

12013 

8.32446 

13787 

7.25310 

9 

52 

08514 

11.7448 

10275 

9.73217 

12042 

8.30406 

13817 

7.23754 

8 

53 

08544 

11.7045 

10305 

9.70441 

12072 

8.28376 

13846 

7.22204 

7 

54 

08573 

11.6645 

10334 

9.67680 

12101 

8.26355 

13876 

7.20661 

6 

55 

08602 

11.6248 

10363 

9.64935 

12131 

8.24345 

13906 

7.19125 

5 

50 

08632 

11.5853 

10393 

9.62205 

12160 

8.22344 

13935 

7.17594 

4 

57 

08661 

11.5461 

10422 

9.59490 

12190 

8.20352 

13965 

7.16071 

3 

58 

08690 

11.5072 

10452 

9.56791 

12219 

8.18370 

13995 

7.14553 

2 

59 

08720 

11.4685 

10481 

9.54106 

12249 

8.16398 

14024 

7.13042 

1 

CO 

08749 

11.4301 

10510 

9.51436 

12278 

8.14435 

14054 

7.11537 

0 

/ 

Cotang 

Tang 

Cotang  | 

Tang1 

Cotang 

Tang 

Cotang 

Tang 

/ 

85° 

84° 

83°            •           82° 

236 


TABLE  XII.-rANUENTS  AND  COTANGENTS.  . 


8° 

9°                        10° 

11° 

f 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

~o 

14054 

7.11537 

15838 

6.31375 

17633 

5.67128 

19438 

5.14455 

60 

1 

14084 

7.10038 

15S68 

6.30189 

17663 

5.66165 

19468 

5.13658 

59 

2 

14113 

7.08546 

15898 

6.29007 

17693 

5.65205 

19498 

5.12862 

58 

3 

14143 

7.07059 

15928 

6.27829 

17723 

5.64248 

19529 

5.12069 

57 

4 

14173 

7.05579 

15958 

6.26655 

17753 

5.63295 

19559 

5.11279 

56 

5 

14202 

7.04105 

15988 

6.25486 

17783 

5.62344 

19589 

5.10490 

55 

6 

14232 

7.02637 

16017 

6.24321 

17813 

5.61397 

19619 

5.09704 

54 

7 

14262 

6.91174 

16047 

6.23160 

17843 

5.60452 

19649 

5.08921 

53 

8 

14291 

6.99718 

16077 

6.22003 

17873 

5.59511 

19680 

5.08139 

52 

9 

14321 

6.98268 

16107 

6.20851 

17903 

5.58573 

19710 

5.07360 

51 

10 

14351 

6.96823 

16137 

6.19703 

17933 

5.57638 

19740 

5.06584 

50 

11 

14381 

6.95385 

16167 

6.18559 

17963 

5.56706 

19770 

5.05809 

49 

12 

14410 

6.93952 

16196 

6.17419 

17993 

5.55777 

19801 

5.05037 

48 

13 

14440 

6.92525 

16226 

6.16283 

18023 

5.54851 

19831 

5.04267 

47 

14 

14470 

6.91104 

16256 

6.15151 

18053 

5.53927 

19861 

5.03499 

46 

15 

14499 

6.89688 

16286 

6.14023 

18083 

5.53007 

19891 

5.02734 

45 

16 

14529 

6.88278 

16316 

6.12899 

18113 

5.52090 

19921 

5.01971 

44 

17 

14559 

6.86874 

16346 

6.11779 

18143 

5.51176 

19952 

5.01210 

43 

18 

14588 

6.85475 

16376 

6.10664 

18173 

5.50264 

5.00451 

42 

19 

14618 

6.84082 

16405 

6.09552 

18203 

5.49356 

20012 

4.99695 

41 

20 

14648 

6.82694 

16435 

6.08444 

18233 

5.48451 

20042 

4.98940 

40 

21 

14678 

6.81312 

16465 

6.07340 

18263 

5.47548 

20073 

4.98188 

39 

14707 

6.79936 

16495 

6.06240 

18293 

5.46648 

20103 

4.97438 

38 

-'•] 

14737 

6.78564 

16525 

6.05143 

18323 

5.45751 

20133 

4.96690 

87 

24 

14767 

6.77199 

16555 

6.04051 

5.44857 

20164 

4.95945 

36 

25 

14796 

6.75838 

16585 

6.02962 

18384 

5.43966 

20194 

4.95201 

35 

26 

14826 

6.74483 

16615 

6.01878 

18414 

5.45077 

20224 

4.94460 

34 

27 

14856 

6.73133 

16645 

6.00797 

18444 

5.42192 

20254 

4.93721 

33 

28 

14886 

6.71789 

16674 

5.99720 

18474 

5.41309 

20285 

4.92984 

32 

29 

11915 

6.70450 

16704 

5.98646 

18504 

5.40429 

20315 

4.92249 

31 

30 

14945 

6.69116 

16734 

5.97576 

18534 

5.39552 

20345 

4.91516 

30 

31 

14975 

6.67787 

16764 

5.96510 

18564 

5.38677 

20376 

4.90785 

29 

32 

15005 

6.66463 

16794 

5.95448 

18594 

5.37805 

20406 

4.90056 

28 

33 

15034 

6.65144 

16824 

5.94390 

18624 

5.36936 

20436 

4.89330 

27 

34 

15064 

6.63831 

16854 

5.93335 

18654 

5.36070 

20466 

4.88605 

26 

35 

15094 

6.62523 

16884 

5.92283 

18684 

5.35206 

20497 

4.87882 

25 

36 

15124 

6.61219 

16914 

5.91236 

18714 

5.34345 

20527 

4.87162 

24 

37 

15153 

6.59921 

16944 

5.90191 

18745 

5.33487 

20557 

4.86444 

23 

88 

15183 

6.58627 

16974 

5.89151 

18775 

5.32631 

20588 

4.85727 

22 

39 

15213 

6.57&39 

17004 

5.88114 

18805 

5.31778 

20618 

4.85013 

21 

40 

15243 

6.56055 

17033 

5.87080 

18835 

5.30928 

20648 

4.84300 

20 

41 

15272 

6.54777 

1     17063 

5.86051 

18865 

5.30080 

20679 

4.83590 

19 

42 

15302 

6.53503 

17093 

5.85024 

18895 

5.29235 

20709 

4.82882 

18 

43 

15332 

6.52234 

17123 

5.84001 

18925 

5.28393 

20739 

4.82175 

17 

44 

15362 

6.50970 

17153 

5.82982 

18955 

5.27553 

20770 

4.81471 

16 

45 

15391 

6.49710 

17183 

5.81966 

18986 

5.26715 

20800 

4.80769 

15 

46 

15421 

6.48456 

17213 

5.80953 

19016 

5.258SO 

20830 

4.80068 

14 

47 

15451 

6.47206 

17243 

5.79944 

19046 

5.25048 

208Q1 

4.79370 

13 

48 

15481 

6.45961 

17273 

5.78938 

19076 

5.24218 

20891 

4.78673 

12 

49 

15511 

6.44720 

17303 

5.77936 

19106 

5.2-3391 

20921 

4.77978 

11 

50 

15540 

6.43484 

17333 

5.76937 

19136 

5.22566 

20952 

4.77286 

10 

51 

15570 

6.42253 

17363 

5.75941 

19166 

5.21744 

20982 

4.76595 

9 

52 

15600 

6.41026 

17393 

5.74949 

19197 

5.20925 

21013 

4.75906 

8 

53 

15630 

6.39804 

17423 

5.73960 

19227 

5.20107 

21043 

4.75219 

7 

54 

15660 

6.38587 

17453 

5.72974 

19257 

5.19293 

21073 

4.74534 

6 

55 

15689 

6.37374 

17483 

5.71992 

19287 

5.18480 

21104 

4.73851 

5 

56 

15719 

6.36165 

17513 

5.71013 

19317 

5.17671 

21134 

4.73170 

4 

57 

15749 

6.34961 

17543 

5.70037 

19347 

5.16863 

21164 

4.72490 

3 

58 

15779 

6.33761 

17573 

5.69064- 

19378 

5.16058 

21195 

4.71813 

2 

59 

15809 

6.32566 

17603 

5.68094 

19408 

5.15256 

21225 

4.71137 

1 

no 

15&38 

6.31375 

17633 

5.67128 

19438 

5.14455 

21256 

4.70463 

0 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

f 

81° 

80° 

79° 

78° 

237 


^w  .**« 

V  OF  THE 

TABLE  'XIL— TANGENTS  AND  COTANGENTS. 


12° 

13° 

14° 

15° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

21256 

4.70463 

^23087 

4.33148 

24933 

4.01078 

26795 

3.73205 

60 

1 

21286 

4.69791 

23117 

4.32573 

24964 

4.00582 

26826 

3.72771 

59 

2 

21316 

4.G9121 

23148 

4.32001 

24995 

4.00086 

26857 

3.72338 

53 

3 

21347 

4.68452 

23179 

4.31430 

25026 

3.99592 

26888 

3.71907 

57 

4 

21377 

4.67786 

23209 

4.30860 

25056 

3.99099 

26920 

3.71476 

56 

5 

21408 

4.67121 

23240 

4.30291 

25087 

3.98607 

26951 

3.71046 

55 

6 

21438 

4.66458 

23271 

4.29724 

25118 

3.98117 

26982 

3.70616 

54 

7 

21469 

4.65797 

23301 

4.29159 

25149 

3.97627 

27013 

3.70188 

53 

8 

21499 

4.65138 

23332 

4.28595 

25180 

3.97139 

27'044 

3.69761 

52 

9 

215S9 

4.G4480 

23363 

4.28032 

25211 

3.9G651 

27076 

3.69335 

51 

10 

21560 

4.63825 

23393 

4.27471 

25242 

3.96165 

27107 

3.68909 

50 

11 

21590 

4.63171 

23424 

4.26911 

25273 

3.95680 

27138 

3.68485 

49 

13 

21621 

4.62518 

23455 

4.26352 

25304 

3.95196 

27169 

3.680G1 

48 

13 

21651 

4.61868 

23485 

4.25795 

25335 

3.94713 

27201 

3.67638 

47 

14 

21683 

4.61219 

23516 

4.25.239 

253G6 

3.94232 

27232 

3.G7217 

4G 

15 

21712 

4.60572 

23547 

4.24685 

25397 

3.93751 

27263 

3.66796 

45 

16 

21743 

4.59927 

23578 

4.24132 

25428 

3.93271 

27294 

3.66376 

44 

17 

21773 

4.59283 

28606 

4.23580 

25459 

3.92793 

27326 

3.65957 

48 

is 

21804 

4.58G41 

23639 

4.23030 

25490 

3.92316 

27357 

3.G5538 

42 

19 

21834 

4.58001 

23670 

4.22481 

25521 

3.91839 

27388 

3.65121 

•11 

20 

21864 

4.57363 

23700 

4.21933 

25552 

3.91364 

27419 

3.64705 

40 

21 

21895 

4.56726 

23731 

4.21387 

25583 

3.90890 

27451 

3.64289 

39 

23 

21925 

4.56091 

23762 

4.20842 

25614 

3,.  90417 

27482 

3.63874 

38 

23 

21956 

4.55458 

23793 

4.20298 

25645 

3\  89945 

27513 

3.63461 

37 

24 

21986 

4.54826 

23823 

4.19756 

25G76 

3.89474 

27545 

3.63048 

86 

23 

22017 

4.54196 

23S54 

4.19215 

25707 

3.89004 

27576 

3.C2G36 

35 

2G 

22047 

4.53568 

23885 

4.18675 

25738 

3.88536 

27607 

3.62224 

34 

27 

22078 

4.52941 

23916 

4.18137 

25769 

3.88068 

27638 

3.61814 

33 

28 

22108 

4.52316 

23946 

4.17600 

25800 

3.87601 

27G70 

3.61405 

32 

29 

22139 

4.51693 

23977 

4.17064 

25831 

3.87136 

27701 

3.G0996 

81 

30 

22169 

4.51071 

24008 

4.16530 

25862 

3.86671 

27732 

3  60588 

30 

31 

22200 

4.50451 

24039 

4.15997 

25893 

3.86208 

27764 

3.60181 

29 

32 

22231 

4.49832 

24069 

4.15465 

25924 

3.85745 

27795 

3.59775 

28 

33 

22261 

4.49215 

24100 

4.14934 

25955 

3.85284 

27826 

3.59370 

27 

34 

22292 

4.48600 

24131 

4.14405 

25986 

3.84S24 

27858 

3.58966 

26 

35 

22322 

4.47986 

24162 

4.13877 

26017 

3.84364 

27889 

3.58562 

;:.-> 

3G 

22353 

4.47374 

24193 

4.13350 

26048 

3.83906 

27921 

3.58160 

24 

37 

22383 

4.46764 

24223 

4.12825 

26079 

3.83449 

27952 

3.57758 

23 

38 

22414 

4.46155 

24254 

4.12301 

26110 

3.82992 

27983 

3.57357 

2-> 

39 

22444 

4.45548 

24285 

4.11778 

26141 

3.82537 

28015 

3.56957 

21 

40 

22475 

4.44942 

24316 

4.11256 

26172 

3.82083 

28046 

3.56557 

20 

41 

22505 

4.44338 

24347 

4.10736 

26203 

3.81630 

28077 

3.56159 

19 

42 

22536 

4.43735 

24377 

4.10216 

26235 

3.81177 

28109 

3.55761 

18 

43 

22567 

4.43134 

24408 

4.09699 

26266 

3.807'26 

28140 

3.55364 

17 

44 

22597 

4.42534 

24439 

4.09182 

26297 

3.80276 

28172 

3.54068 

16 

45 

22628 

4.41936 

24470 

4.08666 

26328 

3.79827 

28203 

3.54573 

15 

46 

22658 

4.41340 

24501 

4.08152 

26359 

3.79378 

28234 

3.54179 

14 

47 

22G89 

4.40745 

24532 

4.07639 

26390 

3.78931 

28266 

3.53785 

13 

48 

22719 

4.40152 

24562 

4.07127 

26421 

3.78485 

28297 

3.53393 

12 

49 

22750 

4.30560 

24593 

4.06G16 

2G452 

3.78040 

28329 

3.53001 

11 

50 

22781 

4.38969 

24624 

4.06107 

26483 

3.77595 

28360 

3.52609 

10 

51 

22811 

4.38381 

24655 

4.05599 

26515 

3.77152 

28391 

3.52219 

G 

52 

22S42 

4.37793 

24G86 

4.05092 

26546 

3.76709 

28423 

3.51829 

8 

53 

22872 

4.37207 

24717 

4.04586 

26577 

3.7G2G8 

28454 

3.51441 

7 

54 

22903 

4.3GG23 

24747 

4.04081 

2GG08 

3.75828 

28486 

3.51053 

6 

55 

22934 

4.3G040 

24778 

4.03578 

2GG39 

3.75388 

28517 

3.506G6 

5 

50 

22964 

4.35459 

24809 

4.03076 

26670 

3.74950 

28549 

3.50279 

4 

57 

22995 

4.34879 

24840 

4.02574 

26701 

3.74512 

28580 

3.49894 

& 

58 

23026 

4.34300 

24871 

4.02074 

26733 

3.74075 

28G12 

3.49509 

2 

59 

23056 

4.33723 

24902 

4.01576 

26764 

3.73G40 

28643 

3.49125 

1 

GO 

23087 

4.33148 

24933 

4.01078 

26795 

3.73205- 

28G75 

3.48741 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

'  Tang 

Cotang 

Tang 

/ 

77° 

76° 

75° 

74° 

238 


TABLE  XII.— TANGENTS  AND  COTANGENTS. 


16° 

17° 

18° 

19° 

f 

/ 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

o 

28675 

3.48741 

30573 

3.270857 

32492 

3.07768 

34433" 

2.90421 

60 

1 

28706 

3.48359 

30605 

3.26745 

32524 

3.07464 

34465 

2.90147 

59 

LJ 

28738 

3.47977 

30637 

3.26406 

32556 

3.07160 

34498 

2.89873 

58 

8 

28769 

3.47596 

30669 

3.26067 

32588 

3.06857 

34530 

2.89600 

57 

4 

28800 

3.47216 

30700 

3.25729 

32621 

3.06554 

34563 

2.89327 

56 

5 

28832 

3.46837 

30732 

3.25392 

32653 

3.06252 

34596 

2.89055 

55 

G 

28864 

3.46458 

30764 

3.25055 

32685 

3.05950 

34628 

2.88783 

54 

y 

28895 

3.46080 

30796 

3.24719 

32717 

3.05649 

34661 

2.88511 

53 

£ 

28927 

3.45703 

30828 

3.24383 

32749 

3.05349 

34693 

2.88240 

52 

9 

28958 

3.45327 

30860 

3.24049 

32782 

3.05049 

34726 

2.87970 

51 

10 

28990 

3.44951 

30891 

3.23714 

32814 

3.04749 

34758 

2  87700 

50 

11 

29021 

3.44576 

30923 

3.23381 

32846 

3.04450 

34791 

2.87430 

49 

12      29053 

3.44202 

30955 

3.23048 

32878 

3.C4152 

34824 

2.87161 

48 

13      29084 

3.43829 

30987 

3.22715 

32911 

3.03854 

34856 

2.86892 

47 

14     29116 

3.43456 

31019 

3.22384 

32943 

3.03556 

34889 

2.86624 

46 

15      29147 

3.43084 

31051 

3.22053 

32975 

3.03260 

34922 

2.86356 

46 

16      29179 

3.42713 

31083 

3.21722 

33007 

3.02963 

34954 

2.86089 

44 

17 

29210 

3.42343 

31115 

3.21392 

33040 

3.02667 

34987 

2.85822 

43 

IS      29242 

3.41973 

31147 

3.21063 

33072 

3.02372 

35020 

2.85555 

42 

19      29274 

3.41604 

31178 

3.20734 

33104 

3.02077 

35052 

2.85289 

41 

20 

29305 

3.41236 

31210 

3.20406 

33136 

3.01783 

35085 

2.85023 

40 

21 

29337 

3.40869 

31242 

3.20079 

33169 

3.01489 

35118 

2.84758 

39 

23 

29368 

3.40502 

31274 

3.19752 

33201 

3.01196 

35150 

2.84494 

38 

X!3 

29400 

3.40136 

31306 

3.19426 

33283 

3.00903 

35183 

2.84229 

37 

94 

29432 

3.39771 

31338 

3.19100 

332G6 

3.00611 

35216 

2.83965 

36 

96 

29403 

3.39406 

31370 

3.18775 

&3298 

3.00319 

35248 

2.83702 

35 

20 

29495 

3.39042 

31402 

3.18451 

33330 

3.00028 

35281 

2.83439 

34 

27 

29526 

3.38679 

31434 

3.18127 

33363 

2.99738 

35314 

2.&3176 

33 

23 

29558 

3.38317 

31466 

3.17804 

33395 

2.99447 

35346 

2.82914 

32 

23 

29590 

3.37955 

31498 

3.17481 

33427 

2.99158 

35379 

2.82653 

31 

80 

29621 

3.37594 

31530 

3.17159 

33460 

2.98868 

35412 

2.82391 

30 

31 

29653 

3.37234 

31562 

3.16838 

33492 

2.98580 

35445 

2.82130 

29 

as 

29685 

3.30875 

31594 

3.16517 

33524 

2.98292 

35477 

2.81870 

^ 

83 

29716 

3.36516 

31626 

3.16197 

33557 

2.98004 

35510 

2.81610 

27 

::4 

29748 

3.30158 

31658 

3.15877 

33589 

2.97717 

35543 

2.81350 

26 

35 

29780 

3.35800 

31690 

3.15558 

33621 

2.97430 

35576 

2.81091 

25 

86 

29811 

3.35443 

31722 

3.15240 

33654 

2.97144 

35608 

2.80833 

24 

37 

29843 

3.35087 

31754 

3.14922 

33686 

2.96858 

35641 

2.80574 

23 

'13 

29875 

3.34732 

31786 

3.14605 

33718 

2.96573 

'35674 

2.80316 

22 

:!9 

29906 

3.34377 

31818 

3.14288 

33751 

2.96288 

35707 

2.80059 

21 

40 

29938 

3.34023 

31850 

3.13972 

33783 

2.96004 

35740 

2.79802 

20 

41 

29970 

3.33670 

31882 

3.13656 

33816 

2.95721 

35772 

2.79545 

19 

42 

30001 

3.33317 

31914 

3.13341 

33848 

2.95437 

35805 

2.79289 

18 

43 

30033 

3.32965 

31946 

3.13027 

33881 

2.95155 

35838 

2.79033 

17 

44 

80065 

3.32614 

31978 

3.12713 

33913 

2.94872 

35871 

2.78778 

16 

45 

30097 

3.32264 

32010 

3.12400 

33945 

2.94591 

35904 

2.78523 

15 

46 

30128 

3.31914 

32042 

3.12087 

33978 

2.94309 

35937 

2.78269 

14 

47 

30160 

3.31565 

32074 

3.11775 

34010 

2.94028 

35969 

2.78014 

13 

48      30192 

3.31216 

32106 

3.11464 

34043 

2.93748 

36002 

2.77761 

12 

49  i     30224 

3.30SG8 

32139 

3.11153 

34075 

2.93468 

36035 

2.77507 

11 

50      30255 

3.30521 

32171 

3.10843 

34108 

2.93189 

36068 

2.77254 

10 

51 

30287 

3.30174 

32203 

3.10532 

34140 

2.92910 

36101 

2.77002 

9 

52 

30319 

3.29829 

32235 

3.10223 

34173 

2.92632 

36134 

2.76750 

8 

53 

30351 

3.29483 

32267 

3.09914 

34205 

2.92354 

36167 

2.76-198 

7 

54 

30382 

3.29139 

32299 

3.09606 

34238 

2.92076 

36199 

2.76247 

6 

56 

30414 

3.28795 

32331 

3.09298 

34270 

2.91799 

36232 

2.75996 

5 

56 

30446 

3.28452 

32363 

3.08991 

34303 

2.91523 

36265 

2.75746 

4 

57 

30478 

3.28109 

32396 

3.08685 

343:35 

2.91246 

36298 

2.75496 

3 

58 

30509 

3.27767 

32428 

3.08379 

34368 

2.90971 

36331 

2.75246 

2 

59 

30541 

3.27426 

32460 

3.08073 

34400 

2.90696 

36364 

2.74997 

1 

GO 

30573 

3.27085 

32492 

3.07768 

34433 

2.90421 

36397 

2.74748 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

[Cotang 

Tang 

Cotang 

Tang 

/ 

73° 

72° 

71° 

i           70° 

239 


TABLE  XII.— TANGENTS  AND  COTANGENTS. 


20° 

21° 

22° 

23° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

r 

0 

36397 

2.74748 

'     38386 

2.60509 

40403 

2.47509 

42447 

2.35585 

GO 

1 

3S430 

2.74499 

38420 

2.60283 

40436 

2.47302 

42482 

2.35395 

59 

2 

36463 

2.74251 

38453 

2.60057 

40470 

2.47095 

42516 

2.35205 

58 

3 

36496 

2.74004 

38487 

2.59831 

40504 

2.46888 

42551 

2.35015 

57 

4 

36529 

2.73756 

38520 

2.59606 

40538 

2.46682 

42585 

2.34825 

56 

5 

36562 

2.73509 

38553 

2.59381 

40572 

2.46476 

42619 

2.34636 

55 

C 

36595 

2.73263 

38587 

2.59156 

40606 

2.46270 

42654 

2.34447 

54 

7 

36628 

2.73017 

38620 

2.58932 

40640 

2.46065 

42GS8 

2.34258 

53 

8 

36661 

2.72771 

38654 

2.58708 

40674 

2.45860 

42722 

2.34069 

52 

9 

36694 

2.72526 

38687 

2.58484 

40707 

2.45655 

42757 

2.33881 

51 

10 

36727 

2.72281 

38721 

2.58261 

40741 

2.45451 

42791 

2.33693 

50 

11 

36760 

8.72036 

38754 

2.58038 

40775 

2.45246 

42826 

2.33505 

49 

12 

36793 

2.71792 

38787 

2.57815 

40809 

2.45043 

428GO 

2.33317 

48 

13 

36826 

2.71548 

38821 

2.57593 

40843 

2.44839 

42894 

2.33130 

47 

11 

36859 

2.71305 

38854 

2.57371 

40877 

2.44636 

42929 

2.32943 

46 

15 

36892 

2.71062 

38888 

2.57150 

40911 

2.44433 

42963 

2.32756 

45 

16 

36925 

2.70819 

38921 

2.56928 

40945 

2.44230 

42998 

2.32570 

44 

17 

36958 

2.70577 

38955 

2.56707 

40979 

2.44027 

43032 

2.32383 

43 

18 

36991 

2.70335 

38988 

2.56487 

41013 

2.43825 

43067 

2.32197 

42 

19 

37024 

2.70094 

39022 

2.56266 

41047 

2.43623 

43101 

2.32012 

41 

20 

37057 

2.69853 

39055 

2.56046 

41081 

2.43422 

43136 

2.31826 

40 

21 

37090 

2.69612 

39089 

2.55827 

41115 

2.43220 

43170 

2.31641 

39 

22 

37123 

2.69371 

39122 

2.55608 

41149 

2.43019 

43205 

2.31456 

33 

23 

37157 

2.69131 

39156 

2.55389 

41183 

2.42819 

43233 

2.31271 

37 

24 

37190 

2.68892 

39190 

2.55170 

41217 

2.42618 

43274 

2.31086 

30 

25 

37223 

9.68653 

39223 

2.54952 

41251 

2.42418 

43308 

2.30902 

35 

26 

37256 

2.68414 

39257 

2.54734 

41285 

2.42218 

43343 

2.30718 

34 

27 

37289 

2.68175 

39290 

2.54516 

41319 

2.42019 

43378 

2.30534 

33 

28 

37322 

2.67937 

39324 

2.54299 

41353 

2.41819 

43412 

2.30351 

32 

29 

37355 

2.67700 

39357 

2.54082 

41387 

2.41620 

43447 

2.30167 

31 

30 

37388 

2.67462 

39391 

2.53865 

41421 

2.41421 

43481 

2.29984 

30 

31 

37422 

2.67225 

39425 

2.53648 

41455 

2.41223 

43516 

2.29801 

29 

32 

37455 

2.66989 

39458 

2.53432 

41490 

2.41025 

43550 

2.29619 

28 

33 

37488 

2.66752 

39492 

2.53217 

41524 

2.40827 

43585 

2.29437 

27 

34 

37521 

2.66516 

39526 

2.53001 

41558 

2.40829 

43620 

2.29254 

26 

35 

37554 

2.66281 

39559 

2.52786 

41592 

2.40432 

43654 

2.29073 

25 

30 

37588 

2.66046 

39593 

2.52571 

41626 

2.40235 

43689 

2.28891 

21 

37 

37621 

2.65811 

39628 

2.52357 

41660 

2.40038 

43724 

2.28710 

23 

38 

37654 

2.65576 

39660 

2.52142 

41694 

2.39841 

43758 

2.28528 

•2'2 

39 

37687 

2.65342 

39694 

2.51959 

41728 

2.39645 

43793 

2.28348    21 

40 

37720 

2.65109 

39727 

2.51715 

41763 

2.39449 

43828 

2.28167 

20 

41 

37754 

2.64875 

39761 

2.51502 

41797 

2.39253 

43862 

2.27987 

19 

42 

37787 

2.64642 

39795 

2.51289 

41831 

2.39058 

43897 

2.27806 

18 

43 

37820 

2.64410 

39829 

2.51076 

418G5 

2.38863 

43932 

2.27626 

17 

44 

37853 

2.64177 

39862 

2.50864 

41899 

2.38668 

43966 

2.27447 

16 

45 

37887 

2.63945 

39S96 

2.50652 

41933 

2.38473 

44001 

2.27267 

15 

46 

37920 

2.63714 

39930 

2.50440 

41968 

2.38279 

44036 

2.27088 

14 

47 

37953 

2.63483 

39963 

2.50229 

42002 

2.38084 

44071 

2.26909 

13 

48 

37986 

2.63252 

39997 

2.50018 

42036 

2.37891 

44105 

2.26730 

12 

49 

38020 

2.63021 

40031 

2.49807 

42070 

2.37697 

44140 

2.26552 

11 

50 

38053 

2.62791 

40065 

2.49597 

42105 

2.37504 

44175 

2.26374 

10 

51 

38086 

2.62561 

40098 

2.49386 

42139 

2.37311 

44210 

3.26196 

9 

52 

38120 

2.62332 

40132 

2.49177 

42173 

2.37118 

44244 

2.26018 

8 

53 

38153 

2.62103 

40166 

2.48967 

42207 

2.36925 

44279 

2.25840 

7 

54 

38186 

2.61874 

40200 

2.48758 

42242 

2.36733 

44314 

2.25663 

G 

55 

38220 

2.61646 

40234 

2.48549 

42276 

2.36541 

44349 

2.25486 

5 

56 

38253 

2.61418 

40267 

2  48340 

42310 

2.36349 

44384 

2.25309 

4 

57 

38286 

2.61190 

40301 

2.48132 

42345 

2.36158 

44418 

2.25132 

3 

58 

38320 

2.60963 

40335 

2.47924 

42379 

2.35967 

44453 

2.24956 

2 

59 

38353 

2.60736 

40369 

2.47716 

42413 

2.35776 

44488 

2.24780 

1 

GO 

38386 

2.  60509 

40403 

2.47509 

42447 

2.35585 

44523 

2.24604 

0 

/ 

Cotang 

"Tang    ' 

Cotang' 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

1           69° 

68° 

67° 

66° 

240 


'TABLE  XII.— TANGENTS   AND   COTANGENTS. 


24° 

25°                       26° 

27° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

44523 

2.24604 

4668T 

2.1445T 

48773 

2.05030 

50953 

1.96261 

GO 

1 

41558 

2.24428 

46GG6 

2.14288 

48809 

2.04879 

50989 

1.96120 

59 

2 

44593 

2.24252 

46702 

2.14125 

48845 

2.04728 

51026 

1.95979 

58 

3 

44627 

2.24077 

46737 

2.139G3 

48881 

2.04577 

51063 

1.95838 

57 

4 

446G2 

2.23902 

46772 

2.13801 

48917 

2.04426 

51099 

1.95698    56 

5 

44G97 

o  03707 

46S08 

2.13639 

48953 

2.04276 

51136 

1.95557  !55 

0 

44732 

2^23553 

46843 

2.13477 

48989 

2.04125 

51173 

1.95417    54 

7 

44767 

2.23378 

46879 

2.13316 

49026 

2.03975 

51209 

1.95277    53 

8 

44802 

2.23204 

46914 

2.13154 

49062 

2.03825 

51246 

1.95137 

52 

9 

44837 

2.23030 

46950 

2.12993 

49098 

2.03675 

51283 

1.94997 

51 

10 

44872 

2.22857 

46985 

2.12833 

49134 

2.03526 

51319 

1.94858 

50 

11 

44907 

2.22683 

47021 

2.12671 

49170 

2.03376 

51356 

1.94718 

49 

18 

44942 

2.22510 

47056 

2.12511 

49206 

2.03227 

51393 

1.94579 

48 

13 

44977 

2.22337 

47092 

2.12350 

49242 

2.03078 

51430 

1.94440 

47 

14 

45012 

2.22164 

47128 

2.12190 

49278 

2.02929 

51467 

1.94301 

46 

15 

45047 

2.21992 

47163 

2.12030 

49315 

2.02780 

51503 

1.94162 

45 

1C 

45082 

2.21819 

47199 

2.11871 

49351 

2.02631 

51540 

1.94023 

14 

17 

45117 

2.21647 

47234 

2.11711 

49387 

2.02483 

51577 

1.93885 

43 

18 

45152 

2.21475 

47270 

2.11552 

49423 

2.02335 

51614 

1.93746 

42 

19 

45187 

2.21304 

47305 

2.11392 

49459 

2.02187 

51G51 

1.93608 

41 

20 

45222 

2.21132 

47341 

2.11233 

49495 

2.02039 

51688 

1.93470 

40 

21 

45257 

2.20961 

47377 

2.11075 

49532 

"2.01891 

51724 

1.93332 

39 

22 

45292 

2.20790 

47412 

2.10916 

49568 

2.01743 

51761 

1.93195 

38 

23 

45327 

2.20619 

47448 

2.10758 

49G04 

2.01596 

51798 

1.93057 

37 

24 

45362 

2.20449 

47483 

2.10600 

49640 

2.01449 

51835 

1.92920 

36 

25 

45397 

2.20278 

47519 

2.10442 

49677 

2.01302 

51872 

1.92782 

35 

26 

45432 

2.20108 

47555 

2.10284 

49713 

2.01155 

51909 

1.92645 

34 

27 

45467 

2.19938 

47590 

2.10126 

49749 

2.01008 

51946 

1.92508 

33 

X 

45502 

2.19769 

47G26 

2.09969 

49786 

2.008G2 

51983 

1.92371 

32 

29 

45538 

2.19599 

47G62 

2.09811 

49822 

2.00715 

52020 

1.92235 

31 

30 

45573 

2.19430 

47698 

2.09654 

49858 

2.00569 

52057 

1.92098 

30 

31 

45608 

2.19261 

47733 

2.09498 

49894 

2.00423 

52094 

1.91962 

29 

32 

45643 

2.19092 

47769 

2.09341 

49931 

2.00277 

52131 

1.91826 

28 

33 

45678 

2.18923 

47805 

2.09184 

49967 

2.00131 

52168 

1.91690 

27 

34 

45713 

2.18755 

47840 

2.09028 

50004 

1.99986 

52205 

1.91554 

2G 

35 

45748 

2.18587 

47876 

2.08872 

50040 

1.99841 

52242 

1.91418 

25 

36 

45784 

2.18419 

47912 

2.08716 

50076 

1.99695 

52279 

1.91282 

24 

37 

45819 

2.18251 

47948 

2.08560 

50113 

1.99550 

52316 

1.91147 

23 

38 

45854 

2.18084 

47984 

2.03405 

50149 

1.99406 

52353 

1.91012 

22 

39 

45889 

2.17916 

48019 

2.08250 

50185 

1.99261 

52390 

1.90876 

21 

40 

45924 

2.17749 

48055 

2.08094 

50222 

1.99116 

52427 

1.90741 

20 

41 

45960 

2.17.582 

48091 

2.07939 

50258 

1.98972 

52464 

1.90607 

19 

42 

45995 

2.17416 

48127 

2.07785 

50295 

1.98828 

52501 

1.90472 

18 

43 

46030 

2.17249 

48163 

2.07G30 

50331 

1.98684 

52538 

1.90337 

17 

44 

46065 

2.17083 

48198 

2.07476 

50368 

1.98540 

52575 

1.90203 

16 

45 

46101 

2.16917 

48234 

2.07321 

50404 

1.98396 

52613 

1.90069 

15 

46 

46136 

2.16751 

48270 

2.07167 

50441 

1.98253 

52650 

1.89935 

14 

47 

46171 

2.16585 

48306 

2.07014 

50477 

1.98110 

52687 

1.89801 

13 

48 

46206 

2.16420 

48342 

2.06860 

50514 

1.97'966 

52724 

1.89667 

12 

49      46242 

2.16255 

48378 

2.06706 

50550 

1.97823 

52761 

1.89533 

11 

50     46277 

2.1G090 

48414 

2.06553 

50587 

1.97681 

52798 

1.89400 

10 

51 

46312 

2.15925 

48450 

2.06400 

50623 

1.97538 

52836 

1.8926t> 

9 

B2 

46348 

2.15760 

48486 

2.06247 

50GGO 

1.97395 

52873 

1.89133 

8 

53 

46383 

2.15596 

48521 

2.06094 

50696 

1.97253 

52910 

1.89000 

7 

54 

46418 

2.15432 

48557 

2.05942 

50733 

1.97111 

52947 

1.88867 

6 

55 

46454 

2.15268 

48593 

2.05790 

50769 

1.96969 

52985 

1.88734 

5 

56 

46489 

2.15104 

48629 

2.05637 

50806 

1.96827 

53022 

1.88602 

4 

57 

46525 

2.14940 

48G65 

2.05485 

50843 

1.96685 

53059 

1.88469 

3 

58 

46560 

2.14777 

4S701 

2.05333 

50879 

1.96544 

53096 

1.88337 

2 

59      46595 

2.14614 

48737 

2.05182 

50916 

1.96402 

53134 

1.88205 

1 

GO 

46631 

2.14451 

48773 

2.05030 

50953 

1.96261 

53171 

1.88073 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

/ 

65° 

64° 

63° 

62° 

241 


TABLE  XII.— TANGENTS  AND  COTANGENTS. 


28° 

29° 

30° 

31° 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

0 

53171 

1.88073 

55431 

1.80405 

57735 

1.73205 

60086 

1.66428" 

60 

1 

53208 

1.87941 

55469 

1.80281 

57774 

1.73089 

60126 

1.66318 

59 

2 

53246 

1.87809 

55507 

1.80158 

57813 

1.72973 

60165 

1.66209 

58 

3 

53283 

1.87677 

55545 

1.80034 

57851 

1.72857 

60205 

1.G6099 

57 

4 

53320 

1.87'546 

55583 

1.79911 

57890 

1.72741 

60245 

1.G5990 

56 

5 

53358 

1.87415 

55621 

1.79788 

57929 

1.72625 

60284 

1.65881 

55 

6 

53395 

1.87283 

55659 

1.79665 

57968 

1.72509 

60324 

1.65772 

54 

7 

53432 

1.87152 

55697 

1.79542 

58007 

1.72393 

60364 

1.65663 

53 

8 

53470 

1.87021 

55736 

1.79419 

58046 

1.72278 

60403 

1.65554 

52 

9 

53507 

1.86891 

55774 

1.79296 

58085 

1.72163 

60443 

1.G5445 

51 

10 

53545 

1.86760 

55812 

1.79174 

58124 

1.72047 

60483 

1.65337 

50 

11 

53582 

1.86630 

55850 

1.79051 

58162 

1.71932 

60522 

1.65228 

49 

12 

53620 

1.86499 

55888 

1.78929 

58201 

1.71817 

60562 

1.65120 

48 

18 

53657 

1.86369 

55926 

1.18807 

58240 

1.71702 

60602 

1.65011 

47 

14 

53694 

1.86239 

55964 

1.78685 

58279 

1.71588 

60642 

1.64903 

46 

15 

53732 

1.86109 

56003 

1.78563 

58318 

1.71473 

60681 

1.64795 

45 

16 

53769 

1.85979 

56041 

1.78441 

58357 

1.71358 

607'21 

1.64687 

44 

17 

53807 

1.85850 

56079 

1.78319 

58896 

1.71244 

60761 

1.64579 

4° 

18 

53844 

1.85720 

56117 

1.78198 

58435 

1.71129 

60801 

1.G4471 

42 

19 

53882 

1.85591 

56156 

1.78077 

58-174 

1.71015 

60841 

1.G4363 

41 

20 

53920 

1.85462 

56194 

1.77955 

58513 

1.70901 

60861 

1.G425G 

40 

21 

53957 

1.85333 

56232 

1.77834 

58552 

1.70787 

GOP21 

1.64148 

39 

22 

53995 

1.85204 

56270 

1.77713 

58591 

1.70673 

60960 

1.64041 

38 

23 

54032 

1.85075 

56309 

1.77592 

58631 

1.70560 

61000 

1.63934 

37 

21 

54070 

1.84946 

561347 

1.77471 

5867'0 

1.70446 

61040 

1.63826 

3G 

25 

54107 

1.84818 

56385 

1.77351 

58709 

1.70332 

61080 

1.63719 

35 

26 

54145 

1.81089 

56424 

1.77230 

58748 

1.70219 

61120 

1.63612 

34 

27 

54183 

1.84561 

56462 

1.77110 

58787 

1.70106 

61160 

1.63505 

33 

28 

54220 

1.84433 

56501 

1.76990 

58826 

1.69992 

61200 

i  leases 

32 

29 

54258 

1.84305 

5G539 

1.76869 

588C5 

1  .C9879 

61240 

1.63292 

31 

30 

54296 

1.84177 

56577 

1.76749 

58905 

1.697G6 

61280 

1.63185 

30 

31 

54333 

1.84049 

56616 

1.76629 

58944 

1.69G53 

61320 

1.G3079 

29 

32 

54371 

1.83922 

56654 

1.76510 

58083 

1.G9541 

61360 

1.62972 

28 

33 

54409 

1.83794 

56693 

1.76390 

59022 

1.69428 

61400 

1.G28G6 

27 

34 

54446 

1.83667 

56731 

1.76271 

59061 

1.69310 

61440 

1.62760 

2G 

35 

54484 

1.83540 

56769 

1.76151 

59101 

1.69203 

61480 

1.62654 

25 

30 

54522 

1.83413 

56808 

1.76032 

59140 

1.69091 

61520 

1.62548 

24 

37 

54560 

1.83286 

56846 

1.75913 

59179 

1.68979 

61561 

1.62442 

23 

38 

54597 

1.83159 

58885 

1.75794 

59218 

1.68866 

61601 

l!  68886 

22 

39 

54635 

1.88033 

56923 

1.75G75 

59258 

1.68754 

61641 

1.62230 

21 

40 

54673 

1.82906 

56962 

1.75556 

59297 

1.68643 

61681 

1.62125 

2U 

41 

54711 

1.82780 

57000 

1.75437 

59336 

1.68531 

61721 

1.62019 

19 

42 

54748 

1.82654 

57039 

1.75319 

59376 

1.68419 

61761 

1.61914 

18 

43 

54786 

1.82528 

57078 

1.75200 

59415 

1.68308 

61801 

1.61808 

17 

44 

54824 

1.82402 

57116 

1.75082 

59454 

1.68196 

61842 

1.61703 

1G 

4* 

54862 

1.82276 

57155 

1.74964 

59494 

1.68085 

61882 

1.61598 

15 

4G 

54900 

1.82150 

57193 

1.74846 

59533 

1.67974 

61922 

1.61493 

14 

47 

54938 

1.82025 

57232 

1.74728 

59573 

1.67863 

61962 

1.61388 

13 

48 

54975 

1.81899 

57271 

1.74G10 

59612 

1.G7752 

62003 

1.61283 

12 

49 

S6013 

1.81774 

57309 

1.74492 

59651 

1.G7641 

62043 

1.61179  ill 

50 

55051 

1.81649 

57348 

1.74375 

59691 

1.07530 

62083 

1.61074 

10 

51 

55089 

1.81524 

57386 

1.74257 

59730 

1.67419 

62124 

1.60970 

9 

52 

55127 

1.81399 

57425 

1.74140 

59770 

1.67309 

62164 

1.60865 

8 

53 

55165 

1.81274 

57464 

1.74022 

59809 

1.67198 

62204 

1.60761 

7 

54 

55203 

1.81150 

57503 

1.73905 

59849 

1.67088 

62245 

1.60657 

6 

55 

55241 

1.81025 

57541 

1.73788 

59888 

1.66978 

62285 

1.60553 

5 

56 

55279 

1.80901 

57580 

1.73671 

59928 

1.66867 

62325 

1.60449 

4 

57 

55317 

1.80777 

57619 

1.73555 

59967 

1.66757 

623G6 

1.60345 

3 

58 

55355 

1.80653 

57657 

1.73438 

60007 

1.66647 

62406 

1.60241 

2 

59 

55393 

1.80529 

57696 

1.73321 

60046 

1.66538 

62446 

1.60137 

1 

60 

55431 

1.80405 

57735 

1.73205 

.60086 

1.66428 

62487 

1.60033 

0 

/ 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

61 

o 

60° 

59' 

68° 

TABLE  XII.— TANGENTS  AND  COTANGENTS. 


32 

o 

33 

o  . 

34° 

35 

0 

Tang     Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang     Cotang 

0 

62487 

1.60033' 

64941 

1.52986 

67451 

1.48256 

70021 

1.42815 

0 

62527 

1.59930 

64982 

1.53888 

67493 

1.48163 

70064 

1.42726 

9 

2 

62568 

1.59826 

65024 

1.53791 

67536 

1.48070 

70107 

1.42638 

8 

3 

62608 

1.59723 

65065 

1.53693 

67578 

1.47977 

70151 

1.42550 

7 

4 

62649 

1.59620 

65106 

1.53595 

67620 

1.47885 

70194" 

1.42462 

6 

62689 

1.59517 

65148 

1.53497 

67G63 

1.47792 

70238 

1.42374 

5 

5 

62730 

1.59414 

65189 

1.53400 

67705 

1.47699 

70281 

1.42286 

4 

7 

62770 

1.59311 

65231 

1.53302 

67748 

1.47607 

70325 

1.42198 

3 

8 

62811 

1.59208 

65272 

1.53205 

67790 

1.47514 

703C8 

1.42110 

2 

9 

62852 

1.59105 

65314 

1.53107 

67832 

1.47422 

70412 

1.42022 

1 

10 

62892 

1.59002 

65355 

1.53010 

67875 

1.47330 

70455 

1.41934 

0 

1 

629S3 

1.58900 

65397 

1.52913 

67917 

1.47238 

70499 

1.41847 

9 

62973 

1.58797 

65438 

1.52816 

G79GO 

1.47146 

70542 

1.41759 

8 

3 

63014 

1.58695 

65480 

1.52719 

68002 

1.47053 

70586 

1.41672 

7 

14 

63055 

1.58593 

65521 

1.52622 

68045 

1.46962 

70629 

1.41584 

6 

15 

63095 

1.58490 

65563 

1.52525 

68088 

1.46870 

70673 

1.41497 

45 

6 

63136 

1.58388 

65G04 

1.52429 

68130 

1.46778 

70717 

1.41409 

44 

63177 

1.58286 

65646 

1.52332 

68173 

1.46686 

707GO 

1.41322 

43 

8 

63217 

1.58184 

65688 

1.52235 

68215 

1.46595 

70804 

1.41235 

0 

19 

63258 

1.58083 

65729 

1.52139 

68258 

1.46503 

70S43 

1.41148 

41 

20 

63299 

1.57981 

65771 

1.52043 

68301 

1.46411 

70891 

1  .41061 

40 

21 

63340 

1.57879 

65813 

1.51946 

68343 

1.46320 

70935 

1.40974 

39 

22 

633SO 

1.57778 

65854 

1.51850 

68386 

1.46229 

70979 

1.40887 

38 

23 

63421 

1.57676 

65896 

1.51754 

68429 

1.46137 

71023 

1.40800 

37 

24 

63462 

1.57575 

65938 

1.51658 

68471 

1.46046 

71066 

1.40714 

36 

25 

63503 

1.57474 

65980 

1.51562 

68514 

1.45955! 

71110 

1.40627 

35 

26 

63544 

1.57372 

66021 

1.51466 

68557 

1.  45864  / 

71154 

1.40540 

•>4 

27 

63584 

1.57271 

66063 

1.51370 

68600 

1.45773 

71198 

1.40454 

53 

28 

63625 

1.57170 

66105 

1.51275 

68642 

1.45682^ 

71242 

1.40367 

.'1:2 

23 

63666 

1.57069 

66147 

1.51179 

68G85 

1.45592 

71285 

1.40281 

31 

30 

63707 

1.56969 

66189 

1.51084 

68728 

1.45501 

71329 

1.40195 

30 

31 

63748 

1.56868 

66230 

1.50988 

68771 

1.45410 

71373 

1.40109 

29 

o 

63789 

1.56767 

66272 

1.50893 

68814 

1.45320 

71417 

1.40022 

a 

33 

63830 

1.56667 

66314 

1.50797 

68857 

1.45229 

71461 

1.89936 

27 

34 

63871 

1.56566 

66356 

1.50702 

68900 

1.45139 

71505 

1.39850 

30 

35 

63912 

1.56466 

66398 

1.50607 

68942 

1.45049 

71549 

1.39764 

25 

36 

63953 

1.56366 

66440 

1.50512 

68985 

1.44958 

71593 

1.39679 

24 

ft 

63994 

1.56265 

66482 

1.50417 

69028 

1.44868 

71637 

1.39593 

03 

oc 

64035 

1.56165 

66524 

1.50322 

69071 

1.44778 

71681 

1.39507 

8 

>9 

64076 

1.56065 

665G6 

1.50228 

69114 

1.44688 

71725 

1.39421 

10 

64117 

1.55966 

66608 

1.50133 

69157 

1.44598 

71769 

1.39336 

20 

41 

€4158 

1.55866 

66650 

1.50038 

69200 

1.44508 

71813 

1.39250 

19 

42 

64199 

1.55766 

66692 

1.49944 

69243 

1.44418 

71857 

1.39165 

18 

43 

64240 

1.55666 

66734 

1.49849 

69286 

1.44329 

71901 

1.39079 

17 

44 

64281 

1.55567 

66776 

1.49755 

69329 

1.44239 

71946 

1.38994 

1G 

45 

64322 

1.55467 

66818 

1.49G61 

69372 

1.44149 

71990 

1.38909 

15 

46 

64363 

1.55363 

66860 

1.49566 

69416 

1.44060 

72034 

1.38824 

14 

47 

64404 

1.55269 

66902 

1.49472 

69459 

1.43970 

72078 

1.38738 

13 

48 

64446 

1.55170 

66944 

1.49378 

69502 

1.43881 

72122 

1.38653 

12 

49 

64487 

1.55071 

66986 

1.49284 

69545 

1.43792 

72167 

1.38568 

11 

50 

64528 

1.54972 

67023 

1.49190 

69588 

1.43703 

72211 

1.38484 

10 

5 

64569 

1.54873 

67071 

1.49097 

69631 

1.43614 

72255 

1.3a399 

9 

5; 

64610 

1.54774 

67113 

1.49003 

69G75 

1.43525 

72299 

1.38314 

8 

5! 

64652 

1.54G75 

67155 

1.48909 

69718 

1.43436 

72344 

1.38229 

7 

54 

64693 

1.54576 

67197 

1.48816 

69761 

1.43347 

72388 

1.38145 

6 

5 

64734 

1.54478 

67239 

1.48722 

69804 

1.43258 

72432 

1.38060 

5 

5 

64775 

1.54379 

87883 

1.48G29 

69847 

1.43169 

72477 

1.37976 

4 

5' 

64817 

1.54281 

67324 

1.48536 

69891 

1.43080 

72521 

1.37891 

3 

5 

64858 

1.54183 

67366 

1.48442 

69934 

1.42992 

72565 

1.37807 

2 

55 

64899 

1.54085 

67409 

1.48349 

69977 

1.42903 

72610 

1.37722 

1 

6_ 

64941 

1.53986 

67451 

1.48256 

70021 

1.42815 

72654  \ 

1.37638 

0 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

Tang 

Cotang 

'  Tang 

f 

57° 

56° 

55° 

54° 

TABLE  XII.— TANGENTS  AND  COTANGENTS. 


36° 

37° 

88° 

39° 

Tang^  Cotang 

Tang     Cotang 

Tang     Cotang 

Tang     Cotang 

/ 

0 

72654      1.37038 

'  75355      1.3-3704 

78129      1.27994 

80978     1.23490 

60 

1 

72699     1.37554 

75401      1.82024 

78175     1.27917 

81027     1.23416 

59 

2 

72743     1.37470 

75447      1.32514 

78222     1.27841 

81075     1.23343 

58 

3 

72788     1.3738G 

75492      1.32464 

78209      1.27764 

81123      1.23270 

57 

4 

72832     1.37302 

75538     1.32384 

78316     1.27688 

81171      1.23190 

66 

5 

72877     1.37218 

75584     1.32304 

78363     1.27611 

81220      1.23123 

55 

C 

7'2921      1.37134 

75629     1.32221 

78410     1.27535 

81268      1.23050 

54 

7 

72966     1.37050 

75675     1.32144 

78457     1.27458 

81316     1.22977 

53 

8 

73010     1.36967 

75721      1.32064 

78504     1.27382 

81364      1.22904 

52 

9 

73055     1.36883 

75767     1.31984 

78551      1.27306 

81413      1.22831 

51 

10 

73100     1.36800 

75812     1.31904 

78598     1.27230 

81461     1.22758 

50 

11 

73144     1.36716 

75858     1.31825 

•78645     1.27153 

81510        .22685 

49 

12 

73189     1.36633 

75904     1.31745 

78092     1.27077 

81558        .22012 

48 

13 

73234      1.36549 

75950     1.31666 

78739     1.27001 

81006        .22539 

47 

14 

73278     1.36466 

75996     1.31586 

78786     1.26925 

81055        .22467 

46 

15 

73323     1.36883 

76042      1.31507 

78834     1.26849 

81703        .22394 

45 

1C 

73368     1.36300 

76088     1.31427 

78881      1.26774 

81752        .22321 

44 

17 

73413     1.36217 

76134     1.31348 

78928     1.26698 

81800        .22249 

43 

18 

73457     1.36134 

76180     1.31269 

78975     1.20622 

81849        .22176 

42 

19 

73502     1.36051 

76226     1.31190 

79022      1.26546 

81898        .22104 

41 

20 

73547     1.35968 

76272     1.31110 

79070     1.26471 

81946        .22031 

40 

21 

73592     1.35885 

76318     1.31031 

79117     1.26395 

81995        .21959 

39 

22 

73637     1.35802 

76364     1.30952 

79104     1.20319 

820  It        .21886 

38 

23 

73681      1.35719 

76410     1.30873 

79212      1.20244 

82092        .21814 

37 

24 

73726     1.35637 

76456     1.30795 

79259     1.20169 

82141        .21742 

36 

25 

73771      1.35554 

76502     1.30716 

79306     1.26093 

82190        .21670 

35 

20 

73816     1.35472 

76548     1.30637 

79354      1.26018 

82238        .21598 

34 

27 

73861      1.35389 

76594      1.30558 

79401      1.25943 

82287        .21526 

33 

28 

73906     1.35307 

76640     1.30480 

79449     1.25807 

82336        .21454 

32 

29 

73951      1.35224 

76686      1.30401 

79496      1.25792 

82385        .21382 

31 

30 

73996     1.35142 

76733     1.30323 

79544     1.25717 

82434       .21310 

30 

31 

74041      1.35060 

76779     1.30244 

79591      1.25642 

82483        .21238 

29 

32 

74086     1.34978 

76825     1.30160 

79039      1.25507 

82531        .21166 

28 

33 

74131      1.34896 

76871      1.30087 

79036      1.25492 

82580        .21094 

27 

34 

74176     1.34814 

76918     1.30009 

79734     1.25417 

82629        .21023 

26 

35 

74221      1.34732 

76964     1.29931 

79781      1.25343 

82678        .20951 

25 

36 

74267     1.34650 

77010     1.29853 

79829     1.25268 

82727       .20879 

24 

37 

74312     1.34568 

77057     1.29775 

79877     1.25193 

82776        .20808 

23 

38 

74357      1.34487 

77103     1.29696 

79924     1.25118 

82825        .20736 

22 

39 

74402     1.34405 

77149     1.29618 

79972     1.25044 

82874       .20665 

21 

40 

74447     1.34323 

77196     1.29541 

80020     1.24969 

82923       .20593 

20 

41 

74492     1.34242 

77242     1.29463 

80067     1.24895 

82972       .20522 

19 

42 

74538     1.34160 

77289     1.29385 

80115      1.24820 

83022        .20451 

18 

43 

74583      1.34079 

77335      1.29307 

80103     1.24746 

83071        .20379 

17 

44 

74628     1.33998 

T382      1.29229 

80211     1.24672 

83120        .20308 

16 

45 

74674     1.33916 

T423      1.29152 

80258     1.24597 

83169        .20237 

15 

46 

74719     1.33835 

T475      1.29074 

80306     1.24523 

83218       .20166 

14 

47 

74764     1.33754 

7521      1.28997 

80354     1.24449 

83268        .20095 

13 

48 

74810     1.33673 

T5G8     1.28919 

80402      1.24375 

83317        .20024 

12 

49 

74855      1.33592 

T615     1.28842 

80450     1.24S01 

83366        .19953 

11 

50 

74900      1.33511 

TGG1     1.28764 

80498     1.24227 

83415        .19882 

10 

51 

74946     1.33430 

77708     1.28687 

80546     1.24153 

83465       .19811 

9 

52 

74991      1.33349 

77754     1.28610 

80594      1.24079 

83514       .19740 

8 

53 

75037     1.33268 

77801      1.28533 

80642      1.24005 

83564        .19669 

7 

54 

75082     1.33187 

77848     1.28456 

80690     1.23931 

83613       .19599 

6 

55 

75128     1.33107 

77895     1.28379 

80738     1.23858 

83662       .19528 

5 

56 

75173     1.33026 

77941      1.28302 

80786     1.23784 

83712        .19457 

4 

57 

75219      1.32946 

77988     1.28225 

808:34     1.23710 

83761        .19387 

3 

58 

75264      1.32865 

78035     1.28148 

80882     1.23637 

83811        .19316 

2 

59 

75310      1.32785 

78082     1.28071 

80930     1.23563 

83860       .19246 

1 

GO 

75355      1.32704 

78129     1.27994 

80978      1.23490 

83910       .19175 

0 

/ 

Cotang     Tang 

Cotang     Tang 

Cotang     Tang 

Cotang     Tang 

/ 

53° 

52° 

51° 

50° 

244 


TABLE  XII.-TANGENTS  AND   COTANGENTS. 


40° 

41° 

42° 

43° 

Tang     Cotang 

Tang     Cotang 

Tang     Cotang 

Tang     Cotang 

o 

83910     1.19175 

86929      1.15037 

90040      1.11061 

93252      1.07237 

GO 

1 

83960      1.19105 

86980      1.14969 

90093     1.10996 

93306     1.07174 

59 

2 

84009      1.19035 

87031      1.14902 

90146      1.10931 

93360     1.07112 

58 

3 

84059     1  .  18964 

87082     1.14834 

90199     1.10867 

93415      1.07049 

57 

4 

84108     1.18894 

87133     1.14767 

90251      1.10802 

93469     1.06987 

50 

5 

84158     1.18824 

87184     1.14699 

90304      1.10737 

93524      1.06925 

55 

c 

84208     1.18754  , 

87236     1.14632 

90357      1.10672 

93578     1.06862 

54 

•7 

84258     1.18684  i 

87287     1.14565 

90410      1.10607 

93633     1.06800 

53 

8 

84307     1.18614  ' 

87338      1.14498 

90463      1.10543 

93688     1.06738 

52 

9 

84357     1.18544 

87389      1.14430 

90516     1.10478 

93742     1.06676 

51 

10 

84407     1.18474 

87441      1.14363 

90569      1.10414 

93797     1.06613 

50 

11 

84457     1.18404 

87492     1.14296 

90621      1.10349 

93852     1.06551 

49 

12 

84507     1.18334 

87543      1.14229 

90674      1.10285 

93906      1.06489 

48 

13 

84556      1.18264 

8^595      1.14162 

90727     1.10220 

93961      1.06-127 

47 

14 

84606     1.18194 

87646     1.14095 

90781      1.10156 

94016      1.00365 

40 

15 

84656     1.18125 

87698     1.14028 

90834     1.10091 

94071      1.06303 

45 

1C 

84706     1.18055 

87749     1.13961 

90887     1.10027 

94125      1.06241 

44 

17 

84756     1.17986 

87801      1.13894 

90940      1.09963 

94180     1.06179 

43 

18 

84806     1.17916 

87852     1.13828 

90993      1.09899 

94235      1.06117 

42 

19 

84856     1.17846 

87904     1.13761 

91046      1.09834 

94290      1.06056 

41 

20 

84906     1.17777 

87955     1.13694 

91099     1.09770 

94345     1.05994 

40 

21 

84956     1.17708 

88007     1.13627 

91153     1.09706 

94400     1.05932 

39 

22 

85006      1.17638 

88059     1.13561 

91206      1.09642 

94455      1.05870 

38 

23 

85057     1.17569 

88110     1.13494 

91259      1.09578 

94510      1.05809 

37 

24 

85107      1.17500 

88162     1.13428 

91313      1.09514 

94565      1.05747 

30 

25 

85157      1.17430 

88214      1.13361 

91366      1.09450 

94620     1.05685 

35 

26 

85207      1.17361 

88265      1.13295 

91419      1.09386 

94676      1.05624 

31 

27 

85257      1.17292 

88317     1.13223 

91473      1.09322 

94731      1.05562 

33 

28 

85308     1.17223 

88369     1.13162 

91526      1.09258 

94786     1.05501 

W 

29 

85358     1.17154 

88421      1.13096 

91580      1.09195 

94841      1.05439 

31 

30 

85408     1.17085 

88473     1.13029 

91633      1.09131 

94896     1.05378 

30 

31 

85458     1.17016 

88524     1.12963 

91687     1.09067 

94952     1.C5317 

29 

32 

85509      1.16947 

88576     1.12897 

91740      1.09003 

95007      1.05255 

28 

33 

85559     1.16878 

88628     1.12831 

91794      1.08940 

95062      1.05194 

27 

34 

85609     1.16809 

88680     1.12765 

91847     1.08876 

95118     1.05133 

20 

35 

85660     1.16741 

88732     1.12699 

91901      1.08813 

95173      1.05072 

« 

38 

85710     1.16672 

88784     1.12633 

91955      1.08749 

95229      1.05010 

21 

37 

85761     1.16603 

88836     1.12567 

92008     1.08686 

95284      1.04949 

23 

38 

85811      1.16535 

88888     1.12501 

92062      1.08622 

95340     1.04888 

22 

39 

85862      1.164JG6 

88940     1.12435 

92116      1.08559 

95395     1.04827 

21 

40 

85912     1.16398 

88992     1.12369 

92170     1.08496 

95451     1.04766 

20 

41 

85963     1.16329 

89045     1.12303 

92224     1.08432 

95506     1.04705 

19 

42 

86014     1.16261 

89097     1.12238 

92277      1.08369 

95562      1.04644 

18 

43 

86064      1.16192 

89149      1.12173 

92331      1.08306 

95618     1.04583 

17 

44 

86115     1.1G124 

89201      1.12106 

92385      1.08243 

95673      1.04522 

16 

45 

86166      1.16056 

89253     1.12041 

92439      1.08179 

95729      1.04461 

15 

46 

86216      1.15987 

89306     1.11975 

92493      1.08116 

95785      1.04401 

14 

47 

86267     1.15919 

89358     1.11909 

92547     1.08053 

95841      1.04340 

13 

48 

86318     1.15851 

89410     1.11844 

92601      1  -07990 

95897      1.04279 

13 

49 

86368     1.15783 

89463     1  11778 

92655      1.07927 

95952      1.04218 

11 

50 

86419     1.15715 

89515     1.11713 

92709     1.07864 

96008     1.04158 

10 

51 

86470     1.15047 

89567     1.11648 

92763     1.07801 

96064     1.04097 

9 

52 

86521      1.15579 

89620     1.11582 

92817     1.07738 

96120     1.04036 

8 

53 

86572     1.15511 

89672     1.11517 

92872     1.07676 

96176      1.03976 

7 

54 

86023      1.15443 

89725      1.11452 

92926     1.07613 

96232     1.03915 

6 

55 

86674      1.15375 

89777     1.11387 

92980     1.07550 

96288     1.03855 

5 

56 

86725      1.15308 

89830     1.11321 

93034     1.07487 

96344     1.03794 

4 

57 

86776     1.15240 

89883     1.11256 

93088      1.07425 

96400     1.03734 

3 

58 

86827     1.15172 

89935     1.11191 

93143     1.07362 

96457      1.0:3674 

2 

59 

86878     1.15104 

89988     1.11126 

93197      1.07299 

96513     1.03613 

1 

60 

86929     1.15037 

90040     1.11061 

93252     1.07237 

96569     1.03553 

0 

/ 

Cotang     Tang 

Cotang     Tang 

Cotang     Tang 

Cotang     Tang 

f 

49° 

48° 

47° 

46° 

245 


TABLE  XII.-TANGENTS  AND  COTANGENTS. 


44° 

44° 

44° 

Tang     Cotang 

Tang    Cotang 

Tang      Cotang 

0 

9G569       .03553 

60 

20 

97700     1.02355 

40 

40 

98843     1.01170 

90 

1 

96625      :  .03493 

59 

21 

97756     1.02295 

39 

41 

98901        .01112 

19 

2 

96681        .03483 

58 

22 

97813      1.02236 

38 

42 

98958        .01053 

18 

8 

96738        .03372 

57 

23 

97870     1.02176 

37 

43 

99016        .00994 

17 

4 

9(5794     :  .03312 

56 

24 

97927      1.02117 

36 

44 

99073        .00935 

Hi 

5 

96850       .03252 

55 

25 

97984      1.02057 

35 

45 

99131        .0087(5 

15 

6 

96907        .03192 

54 

26 

98041      1.01998 

34 

46 

99189        .1.0818 

14 

7 

96063       .03132 

53 

27 

98098     1.01939 

33 

47 

99247     :  .00759 

13 

8 

97020        .03072 

5? 

28 

98155     1.01879 

83 

48 

99304        .00701 

12 

9 

97076       .03012 

51 

29 

98213     1.01820 

31 

49 

99362        .(XW42 

11 

10 

9713JJ        .02952 

50 

30 

98270     1.01761 

30 

50 

99420        .00583 

10 

11 

97189       .02892 

49 

31 

98327     1.01702 

29 

51 

99478        .00525 

9 

12 

97246        .02832 

48 

32 

98384      1.01642 

28 

52 

99536        .00467 

8 

13 

97302       .02772 

47 

33 

98441      1.01583 

27 

53 

99594        .00103 

7 

14 

97359       .02713 

46 

34 

98499     1.01524 

26 

54 

99652        .00350 

6 

15 

97416        .02653 

45 

35 

98556     1.01465 

25 

55 

99710        .00291 

5 

1fl 

97472       .02593 

44 

36 

98613     1.01406 

24 

56 

99768      '  .00233 

4 

1? 

97529        .02533 

43 

37 

98671      1.01347 

23 

57 

99826      :  .00175 

3 

18 

97586        .02474 

42 

38 

98728     1.01288 

22 

58 

99884      :  .00116 

2 

1!) 

97643        .02414 

41 

39 

98786     1.01229 

21 

59 

99942        .00058 

1 

20 

97700        .02355 

40 

40 

98843     1.01170 

20 

60 

1.00000        .00000 

0 

Cotang     Tang 

/ 

/ 

Cotang     Tang 

/ 

/ 

Cotang     Tang 

/ 

45° 

45° 

45° 

246 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


0° 

1° 

2° 

3° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.00000 

.00000 

.00015 

.00015 

.00061 

.00061 

.00137 

.00137 

0 

1 

.00000 

.00000 

.00016 

.00016 

.00062 

.00062 

.00139 

00139 

1 

2 

.00000 

.00000 

.00016 

.00016 

.00063 

.00063 

.00140 

.00140 

2 

3 

.00000 

.00000 

.00017 

.00017 

.00064 

.00064 

.00142 

.00142 

3 

4 

.00000 

.00000 

.00017 

.00017 

.00065 

.00065  , 

.00143 

.00143 

4 

5 

.00000 

.OJOOO 

.00018 

.00018 

.00066 

.00066 

.00145 

.00145 

5 

6 

.00000 

.00000 

.00018 

.00018 

.00067 

.00067 

.00146 

.00147 

6 

7 

.00000 

.00000 

.00019 

.00019 

.00068 

.00068 

.00148 

.00148 

7 

8 

.00000 

.00000 

.00020 

.00020 

.00069 

.00069 

.00150 

.00150 

8 

9 

.00000 

.00000 

.00020 

.00020 

.00070 

.00070 

.00151 

.00151 

9 

10 

.00000 

.00000 

.00021 

.00021 

.00071 

.00072 

.00153 

.00153 

10 

11 

.00001 

.00001 

.00021 

.00021 

.00073 

.00073 

.00154 

.00155 

11 

12 

.00001 

.03001 

.00022 

.00022 

.00074 

.00074 

.00156 

.00156 

12 

13 

.00001 

.00001 

.00023 

.00023 

.00075 

.00075 

.00158 

.00158 

13 

14 

.00001 

.00001 

.00023 

.00023 

.00076 

.00076 

.00159 

.00159 

14 

15 

.00001 

.00001 

.00024 

.00024 

.00077 

.00077 

.00161 

.00161 

15 

16 

.00001 

.00001 

.00024 

.00024 

.00078 

.00078 

.00162 

.00163 

16 

17 

.00001 

.00001 

.00025 

.00025 

.00079 

.00079 

.00164 

.00164 

17 

18 

.00001 

.00001 

.00026 

.00026 

.00081 

.00081 

.00166 

.00166 

18 

19 

.00002 

.00002 

.00025 

.00026 

,00082 

.00082 

.00168 

.00168 

19 

20 

.00002 

:  00002 

.00027 

.00027 

.00083 

.00083 

.00169 

.00169 

20 

21 

.00002 

.00002 

.00028 

.00028 

.00084 

.00084 

.00171 

.00171 

21 

22 

.0000:2 

.00002 

.00023 

.00028 

.00085 

.00085 

.00173 

.00173 

22 

23 

.00002 

.00002 

.00029 

.00029 

.00087 

.00087 

.00174 

.00175 

23 

24 

.00002 

.00002 

.00030 

.00030 

.00088 

.00088 

.00176 

.00176 

24 

25 

.00003 

.00003 

.00031 

.00031 

.00089 

.00089 

.00178 

.00178 

25 

26 

.00003 

.00003 

.00031 

.00031 

.00000 

.00090 

.00179 

.00180 

26 

27 

.00003 

.00003 

.00032 

.00032 

.00091 

.00091 

.00181 

.00182 

27 

28 

.00003 

.00003 

.00033 

.00033 

.00093 

.00093 

.00183 

.00183 

28 

29 

.00004 

.00004 

.00034 

.00034 

.00091 

.00094 

.00185 

.00185 

29 

30 

.00004 

.00004 

.00034 

.00034 

.00003 

.00095 

.00187 

.00187 

30 

31 

.00004 

.00004 

.00035 

.00035 

.00096 

.00097 

.00188 

.00189 

31 

32 

.00004 

.00004 

.00036 

.00036 

.00093 

.00098 

.00190 

.00190 

32 

33 

.00005 

.00005 

.00037 

.00037 

.00039 

.00099 

.00192 

.00192 

33 

34 

.00005 

.00005 

.00037 

.00037 

.00100 

.00100 

.00194 

.00194 

34 

35 

.00005 

.00005 

.00038 

.00038 

.00102 

.00102 

.00196 

.00196 

35 

36 

.00005 

.00005 

.00039 

.00039 

.00103 

.00103 

.00197 

.00193 

33 

37 

.00006 

.00006 

.00040 

.00010 

.00104 

.00104 

.00199 

.00200 

37 

38 

.00006 

.00006 

.00041 

.00041 

.00106 

.00106 

.00201 

.00201 

38 

39 

.00006 

.00006 

.00041 

.00041 

.00107 

.00107  i 

.00203 

.00203 

39 

40 

.00007 

.00007 

.00042 

.00042 

.00108 

.00108 

.00205 

.00205 

40 

41 

.00007 

.00007 

.00043 

.00043 

.0*110 

.00110 

.00207 

.00207 

41 

42 

.00007 

.00007 

.00044 

.00044 

.001H 

.00111 

.00208 

.00203 

42 

43 

.00008 

.00008 

.00045 

.00045 

.00112 

.00113 

.00210 

.00211 

43 

44 

.00008 

.00008 

.00046 

.00046 

.00114 

.00114 

.00212 

.00213 

44 

45 

.00009 

.00009 

.00047 

.00047 

.00115 

.00115 

.00214 

.00215 

45 

46 

.00009 

.00009 

.00048 

.00048 

.00117 

.00117 

.00216 

.00216 

46 

47 

.00009 

.00009 

.00048 

.00048 

.00118 

.00118 

.00218 

.00218 

47 

48 

.00010 

.00010 

.00019 

.00049 

.00119 

.00120 

.00220 

.00220 

48 

49 

.00010 

.00010 

.00050 

.00050 

.00121 

.00121 

.00222 

.00222 

49 

50 

.00011 

.00011 

.00051 

.00051 

.00122 

.00122 

.00224 

.00224 

50 

51 

.00011 

.00011 

.00052 

.00052 

.00124 

.00124 

.00226 

.00226 

51 

52 

.00011 

.00011 

.00053 

.00053 

.00125 

.00125 

.00228 

.00228 

52 

53 

.00012 

.00012 

.00054 

.00054 

.00127 

.00127 

.00230 

.00230 

53 

54 

.00012 

.00012 

.00055 

.00055 

.00128 

.00128 

.00232 

.00232 

54 

55 

.00013 

.00013 

.00056 

.00056 

.00130 

.00130 

.00234 

.00234 

55 

56 

.00013 

.00013 

.00057 

.00057 

.00131 

.00131 

.00236 

.00236 

56 

57 

.00014 

.00014 

.00058 

.00058 

.00133 

.00133 

.00238 

.00238 

57 

58 

.00014 

.00014 

.00059 

.00059 

.00134 

.00134 

.00240 

.00240 

58 

59 

.00015 

.00015 

.00060 

.00060 

.00136 

.00136 

.00242 

.00242 

59 

60 

.00015 

.00015 

.00061 

.00061 

.00137 

.00137 

.00244 

.00244 

60 

247 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


/ 

40 

5C 

6° 

7° 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.00244 

.00244 

~003ST 

.00382 

.00548 

.00551 

.00745 

.00751 

0 

1 

.00246 

.00246 

.00383 

.00385 

.00551 

.00554 

.00749 

.00755 

1 

2 

.00248 

.00248 

.00386 

.00387 

.00554 

.00557 

.00752 

.00758 

2 

3 

.00250 

.00250 

.00388 

.00390 

.00557 

.00560 

.00756 

.00762 

3 

4 

.00252 

.00252 

.00391 

.00392 

.00560 

.00563 

.00760 

.00765 

4 

5 

.00254 

.00254 

.00393 

.00395 

.00563 

.00566 

.00763 

.00769 

5 

6 

.00256 

.00257 

.00396 

.00397 

.00566 

.00569 

.00767 

.00773 

6 

7 

.00258 

.00259 

.00398 

.00400 

.00569 

.00573 

.00770 

.00776 

7 

8 

.00260 

.00261 

.00401 

.00403 

.00572 

.00576 

.00774 

.00780 

8 

*  9 

.00262 

.00263 

.00404 

.00405 

.00576 

.00579 

.00778 

.00784 

9 

10 

.00264 

.00265 

.00406 

.00408 

.00579 

.00582 

.00781 

.00787 

10 

11 

.00266 

.00267 

.00409 

.00411 

.00582 

.00585 

.00785 

.00791 

11 

12 

.00269 

.00269 

.00412 

.00413 

.00585 

.00588 

.00789 

.00795 

12 

13 

.00271 

.00271 

.00414 

.00416 

.00588 

.00592 

.00792 

.00799 

13 

14 

.00273 

.00274 

.00417 

.00419 

.00591 

.00595 

.00796 

.00802 

14 

15 

.00275 

.00276 

.00420 

.00421 

.00594 

.  .00598 

.00800 

.00806 

15 

16 

.00277 

.00278 

.00422 

.00424 

.00598 

.00601 

.00803 

.00810 

16 

17 

.00279 

.00280 

.00425 

.00427 

.00601 

.00604 

.00807 

.00813 

17 

18 

.00281 

.00282 

.00428 

.00429 

.00604 

.00608 

.00811 

.00817 

18 

19 

.00284 

.00284 

.00430 

.00432 

.00607 

.00611 

.00814 

.00821 

19 

20 

.00286 

.00287 

.00433 

.00435 

.00610 

.00614 

.00818 

.00825 

20 

21 

.00288 

.00289 

.00436 

.00438 

.00614 

.00617 

.00822 

.00828 

21 

22 

.00290 

.00291 

.00438 

.00440 

.00617 

.00621 

.00825 

.00832 

22 

23 

.00293 

.00293 

.00441 

.00443 

.00620 

.00624 

.00829 

.00836 

23 

24 

.00295 

.00296 

.00444 

.00446 

.00623 

.00627 

.00833 

.00840 

24 

25 

.00297 

.00298 

.00447 

.00449 

.00626 

.00630 

.00837 

.00844 

25 

26 

.00299 

.00300 

.00449 

.00151 

.OOG30 

.00634 

.00840 

.00848 

26 

27 

.00301 

.00302 

.00452 

.00454 

.00633 

.00637 

.00844 

.00851 

27 

28 

.00304 

.00305 

.00455 

.00457 

.00636 

.00640 

.00848 

.00855 

28 

29 

.00306 

.00307 

.00458 

.00460 

.00640 

.00644 

.00852 

.00859 

29 

30 

.00308 

.00309 

.00460 

.00463 

.00643 

.00647 

.00856 

.00863 

30 

31 

.00311 

.00312 

.00463 

.00465 

.00646 

.00650 

.00859 

.00867 

31 

32 

.00313 

.00314 

.00466 

.00468 

.00649 

.00654 

.00863 

.00871 

32 

33 

.00315 

.00316 

.00469 

.00471 

.00653 

.00657 

.00867 

.00875 

33 

34 

.00317 

.00318 

.00472 

.00474 

.00656 

.00660 

.00871 

.00878 

34 

35 

.00320 

.00321 

.00474 

.00477 

.00059 

.00664 

.00875 

.00882 

35 

36 

.00322 

.00323 

.00477 

.00480 

.00663 

.00667 

.00878 

.00886 

36 

37 

.00324 

.00326 

.00480 

.00482 

.00666 

.00671 

.00882 

.00890 

37 

38 

.00327 

.00328 

.00483 

.00485 

.00669 

.00674 

.00886 

.00894 

38 

39 

.00329 

.00330 

.00486 

.00488 

.00673 

.00677 

.00890 

.00898 

39 

40 

.00332 

.00333 

.00489 

.00491 

.00676 

.00681 

.00894 

.00902 

40 

41 

.00334 

.00335 

.00492 

.00494 

.00680 

.00684 

.00898 

.00906 

41 

42 

.00336 

.00337 

.00494 

.00497 

.00688 

.00688 

.00902 

.00910 

42 

43 

.00339 

.00340 

.00497 

.00500 

.OOG86 

.00091 

.00906 

.00914 

43 

44 

.00341 

.00342 

.00500 

.00503 

.00690 

.00695 

.00909 

.00918 

44 

45 

.00343 

.00345 

.00503 

.00506 

.00693 

.00098 

.00913 

.00922 

45 

46 

.00346 

.00347 

.00506 

.00509 

.00697 

.00701  I 

.00917 

.00926 

46 

47 

.00348 

.00350 

.00509 

.00512 

.00700 

.00705  1 

.00921 

.00930 

47 

48 

.00351 

.00352 

.00512 

.00515 

.00703 

.00708 

.00925 

.00934 

48 

49 

.00353 

.00354 

.00515 

.00518 

.00707 

.00712 

.00929 

.00938 

49 

50 

.00356 

.00357 

.00518 

.00521 

.00710 

.00715 

.00933 

.00942 

50 

51 

.00.358 

.00359 

.00521 

.00524 

.00714 

.00719 

.00937 

.00946 

51 

52 

.00361 

.00362 

.00524 

.00527 

.00717 

.00722 

.00941 

.00950 

52 

53 

.00363 

.00364 

.00527 

.00530 

.00721 

.00726 

.00945 

.00954 

53 

54 

.00365 

.00367 

.00530 

.00533 

.00724 

.00730 

.00949 

.00958 

54 

55 

.00368 

.00369 

.00533 

.00536 

.00728 

.00733 

.00953 

.00982 

55 

56 

.00370 

.00372 

.00536 

.00539 

.00731 

.00737 

.00957 

.00066 

56 

57 

.00373 

.00374 

.00539 

.00542 

.00735 

.00740 

.00961 

.00970 

57 

58 

.00375 

.00377 

.00542 

.00545 

.00738 

.00744 

.00965 

.00975 

58 

59 

.00378 

.00379 

.00545 

.00548 

.00742 

.00747 

.00969 

.00979 

59 

60 

.00381 

.00382 

.00548 

.00551 

.00745 

.00751 

.00973 

.00983  1  60 

248 


TABLE  XIIL— VERSINES  AND  EXSECANTS. 


8° 

9° 

10° 

11° 

i 

Vers. 

Exsec. 

Vers. 

£xsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.00973 

.00983 

.01231 

.01247 

.01519 

.01543  1 

.01837 

.01872 

0 

1 

.00977 

.00987 

.01236 

.01251 

.01524 

.01548 

.01843 

.01877 

1 

2 

00981 

.00991 

.01240 

.01256 

.01529 

.01553 

.01848 

.01883 

2 

3 

.00985 

.00995 

.01245 

.01201 

.01534 

.01558 

.01854 

.01889 

3 

4 

.00989 

.00999 

.01249 

.01265 

.01540 

.01564 

.01860 

.01895 

4 

5 

.00994 

.01004 

.01254 

.01270 

.01545 

.01509 

.01865 

.01901 

5 

6 

.00998 

.01008 

.01259 

.01275 

.01550 

.01574 

.01871 

.01906 

6 

7 

01002 

.01012 

.01263 

.01279 

.01555 

.01579 

.01876 

.01912 

7 

8 

.01006 

.01016 

.01268 

.01284 

.01560 

.01585 

.01882 

.01918 

8 

9 

.01010 

.01020 

.01272 

.01289 

.01565 

.01590 

.01888 

.01924 

9 

10 

.01014 

.01024 

.01277 

.01394 

.01570 

.01595 

.01893 

.01930 

10 

11 

.01018 

.01029 

.01282 

.01298 

.01575 

.01001 

.01899 

.01936 

11 

12 

.01022 

.01033 

.01286 

.01303 

.01580 

.01006 

.01004 

.01941 

12 

13 

.01027 

.01037 

.01291 

.01308 

.01586 

.01611 

.01910 

.01947 

13 

14 

.01031 

.01041 

.01296 

.01313 

.01591 

.01616 

.01916 

.01953 

14 

15 

.01035 

.01046 

.01300 

.01318 

.01596 

.01622 

.01921 

.01959 

15 

16 

.01039 

.01050 

.01305 

.01322 

.01601 

.01027 

.01927 

.01965 

16 

17 

.01043 

.01054 

.01310 

.01327 

.01606 

.01033 

.01933 

.01971 

17 

18 

.01047 

.01059 

.01314 

.01332 

.01612 

.01038 

.01939 

.01977 

18 

19 

.01052 

.01063 

.01319 

.01337 

.01617 

.01043 

.01944 

.01983 

19 

20 

.01056 

.01067 

.01324 

.01342 

.01622 

.01649 

.01950 

.01989 

20 

21 

.01060 

.01071 

.01329 

.01346 

.01627 

.01654 

.01956 

.01995 

21 

22 

.01064 

.01076 

.013:33 

.01351 

.01032 

.01059 

.01961 

.02001 

22 

23 

.010G9 

.01080 

.01338 

.01356 

.  .01638 

.01665 

.01967 

.02007 

23 

24 

.01073 

.01084 

.01343 

.01361 

.01643 

.01070 

.01973 

.02013 

24 

25 

.01077 

.01089 

.01348 

.01366 

.01648 

.01676 

.01979 

.02019 

25 

23 

.01081 

.01093 

.01352 

.01371 

.01653 

.01681 

.01984 

.02025 

26 

27 

.01086 

.01097 

.01357 

.01376 

.01659 

.01687 

.01990 

.02031 

27 

28 

.01090 

.01102 

.01362 

.01381 

.01064 

.01692 

.01996 

.02037 

28 

29 

.01094 

.01106 

.01307 

.01386 

.01009 

.01698 

.02002 

.02043 

29 

30 

.01098 

.01111 

..01371 

.01391 

.01675 

.01703 

.02008 

.02049 

30 

31 

.01103 

.01115 

.01376 

.01395 

.01680 

.01709 

.02013 

.02055 

31 

32 

.01107 

.01119 

.01381 

.01400 

.01085 

.01714 

.02019 

.02001 

33 

33 

.01111 

.01124 

.01386 

.01405 

.01690 

.01720 

.02025 

.02007 

33 

34 

.01116 

.01128 

.01391 

.01410 

.01096 

.01725 

.02031 

.02073 

31 

35 

.01120 

.01133 

.01396 

.01415 

.01701 

.01731 

.02037 

.02079 

35 

36 

.01124 

.01137 

.01400 

.01420 

.01706 

.01736 

.02042 

.02085 

36 

37 

.01129 

.01142 

.01405 

.01425 

.01712 

.01742 

.02048 

.02091 

37 

38 

.01133 

.01146 

.01410 

.01430 

.01717 

.01747 

.02054 

.02097 

38 

39 

.01137 

.01151 

.01415 

.014-35 

.01723 

.01753 

.020GO 

.02103 

39 

43 

.01142 

.01155 

.01420 

.01440 

.01728 

.01758 

.02066 

.02110 

40 

41 

.01146 

.01160 

.01425 

.01445 

.01733 

.01764 

.02072 

.02116 

41 

42 

.01151 

.01164 

.01430 

.01450 

.01739 

.01769 

.02078 

.02122 

42 

43 

.01155 

.01169 

.01435 

.01455 

.01744 

.01775 

.02084 

.02128 

43 

44 

.01159 

.01173 

.01439 

.01401 

.01750 

.01781 

.02090 

.02134 

44 

45 

.01164 

.01178 

.01444 

.01406 

.01755 

.01786 

.02095 

.02140 

45 

46 

.01168 

.01182 

.01449 

.01471 

.01760 

.01792 

.02101 

.02146 

46 

47 

.01173 

.01187 

.01454 

•01476 

.01766 

.01798 

.02107 

.02153 

47 

48 

.01177 

.01191 

.01459 

.01481 

.01771 

.01803 

.02113 

.02159 

48 

49 

.01182 

.01196 

.01464 

.01486 

.01777 

.01809 

.02119 

.02165 

49 

50 

.01186 

.01200 

.01469 

.01491 

.01782 

.01815 

.02125 

.02171 

50 

51 

.01191 

.01205 

.01474 

.01496 

.01788 

.01820 

.02131 

.02178 

51 

52 

.01195 

.01209 

.01479 

.01501 

.01793 

.01826 

.02137 

.02184 

52 

53 

.01200 

.01214 

.01484 

.01506 

.01799 

.01832 

.02143 

.02190 

53 

54 

.01204 

.01219 

.01489 

.01512 

.01804 

.01837 

.02149 

.02196 

54 

55 

.01209 

.01223 

.01494 

.01517 

.01810 

.01843 

.02155 

.02203 

55 

56 

.01213 

.01228 

.01499 

.01522 

.01815 

.01849 

.02161 

.02209 

56 

57 

.01218 

.01233 

.01504 

.01527 

.01821 

.01854 

.02167 

.02215 

57 

58 

.01222 

.01237 

.01509 

.01532 

.01826 

.01860 

.02173 

.02221 

58 

59 

.01227 

.01242 

.01514 

.01537 

.01832 

.01866 

.02179 

.02228 

59 

60 

.01231 

.01247 

.01519 

.01543 

.01837 

.01872 

.02185 

.02234 

60 

249 


TABLE  XIII.-VERSINES  AND  EXSECANTS. 


/ 

12° 

13° 

14° 

15° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

~0~ 

.02185 

.02234 

.02563 

.02630 

.02970 

.03061 

.03407 

.03528 

0 

1 

.02191 

.02240 

.02570 

.02637 

.02977 

.03069 

.03415 

.03536 

1 

2 

.02197 

.02247 

.02576 

.02644 

.02985 

.03076 

.03422 

.03544 

2 

3 

.02203 

.02253 

.02583 

.02651 

.02992 

.03084 

.03430 

.03552 

3 

4 

.02210 

.02259 

.02589 

.02658 

.02999 

.03091 

.03438 

.03560 

4 

5 

.02216 

.02266 

.02596 

.02665 

.03006 

.03099 

.03445 

.035G8 

5 

6 

.02222 

.02272 

.02602 

.02(372 

.03013 

.03106 

.03453 

.03576 

6 

7 

.02228 

.02279 

.02609 

.02679 

.03020 

.03114 

.03460 

.03584 

7 

8 

.02234 

.02285 

.02616 

.C2686 

.03027 

.03121 

.03468 

.03592 

8 

9 

.02240 

.02291 

.02622 

.02693 

.03034 

.03129 

.03476 

.03601 

9 

10 

.02246 

.02298 

.02629 

.02700 

.03041 

.03137 

.03483 

.03609 

10 

11 

.02252 

.02304 

.02635 

.02707 

.03048 

.03144 

.03491 

.03617 

11 

12 

.02258 

.02311 

.02642 

.02714 

.03055 

.03152 

.03-498 

.03625 

12 

13 

.02265 

.02317 

.02649 

.02721 

.03063 

.03159 

.03506 

.03633 

13 

14 

.02271 

.02323 

.02655 

.02728 

.03070 

.03167 

.03514 

.03642 

14 

15 

.02277 

.02330 

.02662 

.02735 

.03077 

.03175 

.03521 

.03650 

15 

16 

.02233 

.02336 

.02669 

.02742 

.03084 

.03182 

.03529 

.03658 

16 

17 

.02289 

.02343 

.02675 

.02749 

.03091 

.03190 

.03537 

.03666 

17 

18 

.02295 

.02349 

.02682 

.02756 

.03098 

.03198 

.03544 

.03674 

18 

19 

.02302 

.02356 

.02689 

.02763 

.03106 

.03205 

.03552 

.03683 

19 

20 

.02308 

.02362 

.02696 

.02770 

.03113 

.03213 

.03560 

.03691 

20 

21 

.02314 

.02369 

.02702 

.02777 

.03120 

.03221 

.03567 

.03699 

21 

22 

.02320 

.02375 

.02709 

.02784 

.03127 

.03228 

.0357'5 

.03708 

22 

23 

.02327 

.02382 

.02716 

.02791 

.03134 

.03236 

.03583 

.03716 

23 

24 

.02333 

.02388 

.02722 

.02799 

.03142 

.03244 

•  .03590 

.037'24 

24 

25 

.02339 

.02395 

.02729 

.02806 

.03149 

.03251 

.03598 

.03732 

25 

'  26 

.02345 

.C2402 

.02736 

.02813 

.03156 

.03259 

.03606 

.03741 

2(3 

27 

.02352 

.02408 

.02743 

.02820 

.03163 

.03267 

.03614 

.03749 

27 

28 

.02358 

.02415 

.02749 

.02827 

.03171 

.03275 

.03621 

.03758 

28 

£3 

.02364 

.02421 

.02756 

.02834 

.03178 

.03282 

,,03629 

.03766 

29 

SO 

.02370 

.02428 

.02763 

.02843 

.03185 

.03290 

.03637 

.03774 

30 

31 

.02377 

.02435 

.02770 

.02849 

.03193 

.03298 

.03645 

.03783 

31 

S3 

.02383 

.02441 

.02777 

.02856 

.03200 

.03306 

.03653 

.03791 

32 

S3 

.02389 

.02448 

.02783 

.028G3 

.03207 

.03313 

.03660 

.03799 

33 

34 

.02396 

.02454 

.02790 

.02870 

.03214 

.03321 

.03608 

.03808 

34 

35 

.02-402 

.02461 

.02797 

.02878 

.03222 

.03329 

.03676 

.03816 

35 

36 

.02408 

.02468 

.02804 

.02885 

.03229 

.03337 

.03684 

.03825 

36 

37 

.02415 

.02474 

.02811 

.02892 

.03236 

.03345 

.03692 

.03833 

37 

38 

.02421 

.02481 

.02818 

.02899 

.03244 

.03353 

.03699 

.03842 

38 

^Q 

.02427 

.02488 

.02824 

.02907 

.03251 

.03360 

.03707 

.03850 

W 

40 

.02434 

.02494 

.02831 

.02914 

.03258 

.03368 

.03715 

.03858 

4d 

41 

.02440 

.02501 

.02838 

.02921 

.03266 

.03376 

.03723 

.03867 

41 

42 

.02447 

.02508 

.02845 

.02928 

.03273 

.03384 

.03731 

.0387'5 

42 

43 

.02453 

.02515 

.02852 

.02936 

.03281 

.03392 

.03739 

.03884 

43 

44 

.02459 

.02521 

.02859 

.02943 

.03288 

.03400 

.03747 

.03892 

44 

45 

.02466 

.02528 

.02866 

.02950 

.03295 

.03408 

.03754 

.03901 

45 

46 

.02472 

.02535 

.02873 

.02958 

.03303 

.03416 

.08762 

.03909 

46 

47 

.02479 

.02542 

.02880 

.02965 

.03310 

.03424 

.03770 

.03918 

47 

48 

.02485 

.02548 

.02887 

.02'J7'2 

.03318 

.03432 

.03778 

.03927 

48 

49 

.02492 

.02555 

.02894 

.02980 

.03325 

.03439 

.03786 

.03035 

49 

50 

.0^98 

.02562 

.02900 

.02987 

.03333 

.03447 

.03794 

.03944 

50 

51 

.02504 

.02569 

.02907 

.02994 

.03340 

.0345E 

.03802 

.03952 

51 

52 

.02511 

.02576 

.02914 

.03002 

.03347 

.03463 

.03810 

.03961 

52 

53 

.02517 

.02582 

.02921 

.03009 

.03355 

.03471 

.03818 

.03969 

53 

54 

.02524 

.02589 

.02928 

.03017 

.03362 

.03479 

.03826 

.03978 

54 

55 

.02530 

.02596 

.02935 

.03024 

.03370 

.03487 

.03834 

.03987 

55 

56 

.02537 

.02603 

.02942 

.03032 

.03377 

.03495 

.03842 

.03995 

56 

57 

.02543 

.02610 

.02949 

.03039 

.03385 

.03503 

.03850 

.04004 

57 

58 

.02550 

.02617 

.02956 

.03046 

.03392 

.03512 

.03858 

.04013 

58 

59 

.02556 

.02624 

.02963 

.03054 

.03-100 

.03520 

.03866 

.04021 

59 

60 

.02563 

.02630 

.02970 

.03061 

.03407 

.03528 

.03874 

.04030 

60 

250 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


/ 

16° 

17° 

18" 

19' 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.03874 

.04030 

.04370 

.04569 

.04894 

.05146 

.05448 

.05762 

0 

1 

.03882 

.04039 

.04378 

.04578 

.04903 

.05156 

.05458 

.05773 

1 

2 

.03890 

.04047 

.04387 

.04588 

.04912 

.05166 

.05467 

.05783 

2 

3 

.03898 

.04056 

.04395 

.04597 

.04921 

.05176 

.05477 

.05794 

3 

4 

.03906 

.04065 

.04404 

.04606 

.04930 

.05186 

.65486 

.05805 

4 

5 

.03914 

.04073 

.04412 

.04616 

.04939 

.05196 

.05496 

.05815 

5 

6 

.03922 

.04082 

.04421 

.04625 

.04948 

.05206 

.05505 

.05826 

6 

7 

.03930 

.04091  ! 

.04429 

.04635 

.04957 

.65216 

.05515 

.05836 

7 

8 

.03938 

.04100  | 

.04438 

.04644 

.04967 

.05226  ! 

.05524 

.05847 

8 

g 

.03946 

.04108 

.04446 

.04653 

.04976 

.05236 

.05534 

.05858 

9 

10 

.03954 

.04117  | 

.04455 

.04663 

.04985 

.05246 

.05543 

.05869 

10 

11 

.03963 

.04126 

.04464 

.04672 

.04994 

.05256 

.05553 

.05879 

11 

13 

.03971 

.04135  1 

.04472 

.04683 

.05003 

.05266 

.05562 

.05890 

12 

13 

.03979 

.04144  ! 

.04481 

.04691 

.05012 

.05276 

.05572 

.05901 

13 

14 

.03987 

.04152  ' 

.04489 

.04700 

.05021 

.05286 

.05582 

.05911 

14 

15 

.03995 

.04161  i 

.04498 

.04710  [ 

.05030 

.05297 

.05591 

.05922 

15 

16 

.01003 

.04170 

.04507 

.04719  i 

.05039 

.05307 

.05601 

.05933 

16 

17 

.04011 

.04179 

.04515 

.04729 

.05048 

.05317 

.05610 

.05944 

17 

18 

.01019 

.04188 

.04524 

.04738 

.05057 

.05327 

.05620 

.05955 

18 

19 

.04028 

.04197 

.04533 

.04748 

.05067 

.05337  i 

.05630 

.05965 

19 

20 

.04036 

.04206 

.04541 

.04757 

.05076 

.05347 

.05639 

.05976 

20 

21 

.04044 

.04214 

.04550 

.04767 

.05085 

.05357 

.05649 

.05987 

21 

22 

.04052 

.04223 

.04559 

.04776 

.05094 

.05367 

.05658 

.05998 

22 

23 

.04060 

.04232 

.04567 

.04786 

.05103 

.05378 

.05668 

.06009 

23 

24 

.04069 

.04241 

.04576 

.04795 

.05112 

.05388 

.05678 

.06020 

24 

25 

.04077 

.04250  | 

.04585 

.04805 

.05122 

.05398 

.05687 

.06030 

25 

26 

.04085 

.04259  ! 

.04593 

.04815 

.05131 

.05408 

.05697 

.06041 

26 

27 

.04093 

.04268 

.04602 

.04824 

.05140 

.05418 

.05707 

.06052 

27 

28 

.04102 

.04277 

.04611 

.04834 

.05149 

.05429 

.05716 

.06063 

28 

29 

.04110 

.04286 

.04020 

.04843 

.05158 

.05439 

.05726 

.06074 

29 

30 

.04118 

.04295 

.04628 

.04853 

.05168 

.05449 

.05736 

.06085 

30 

31 

.04126 

.04304 

.04637 

.04863 

.05177 

.05460 

.05746 

.06096 

31 

32 

.04ia5 

.04313 

.04646 

.04872 

.05186 

.05470 

.05755 

.06107 

82 

33 

.04143 

.04322 

.04655 

.04882 

.05195 

.05480 

.05765 

.06118 

33 

34 

.04151 

.04331 

.04663 

.04891 

.05205 

.05490 

.05775 

.06129 

34 

35 

.04159 

.04340 

.04672 

.04901 

.05214 

.05501 

.05785 

.06140 

35 

36 

.04168 

.04349 

.04681 

.04911 

.05223 

.05511 

.05794 

.06151 

36 

37 

.04176 

.04358 

.04690 

.04920 

.05232 

.05521 

.05804 

.06162 

37 

38 

.04184 

.04367 

.04699 

.04930 

.05242 

.05532 

.05814 

.06173 

38 

39 

.04193 

.04376 

.04707 

.04940 

.05251 

.C5542 

.05824 

.06184 

39 

40 

.01201 

.04385 

.04716 

.04950 

.05260 

.05552 

.05833 

.06195 

40 

41 

.04209 

.04394 

.01725 

.04959 

.05270 

.05563 

.05843 

.06206 

41 

42 

.01218 

.04403 

[04784 

.04969 

.05279 

.05573 

.05853 

.06217 

42 

43 

.04226 

.04413 

.04743 

.04979 

.05288 

.05584 

.05863 

.06228 

43 

44 

.04234 

.04422 

.04752 

.04989 

.05298 

.05594 

.05873 

.06239 

44 

45 

.04243 

.04431 

.04760 

.04998 

.05307 

.05604 

.05882 

.06250 

45 

46 

.04251 

.04440 

.04769 

.05008 

.05316 

.05615 

.05892 

.06261 

46 

47 

.04260 

.04449 

.04778 

.05018 

.05326 

.05625 

.05902 

.06272 

47 

48 

.04268 

.04458 

.04787 

.05028 

.05335 

.05636 

.05912 

.06283 

48 

49 

.04276 

.04468 

.04796 

.05038 

.05344 

.05646 

.05922 

.06295 

49 

50 

.04285 

.04477 

.04805 

.05047 

.05354 

.05057 

.05932 

.06306 

50 

51 

.04293 

.04486 

.04814 

.05057 

.05363 

.05667 

.05942 

.06317 

51 

52 

.04302 

.04495 

.04823 

.05067 

.05373 

.05678 

.05951 

.06328 

52 

53 

.04310 

.04504 

.04832 

.05077 

.05382 

.05688 

.05961 

.06339 

53 

54 

.04319 

.04514 

.04841 

.05087  i  .05391 

.05699 

.05971 

.06350 

54 

55 

.04327 

.04523 

.04850 

.05097  |l  .05401 

.05709 

.05981 

.06362 

55 

56 

.04336 

.04532 

.04858 

.05107 

.05410 

.05720 

.05991 

.06373 

56 

57 

.04344 

.04541 

.04867 

.05116 

.05420 

.05730 

.06001 

.06384 

57 

58 

.04&53 

.04551 

.04876 

.05126 

.05429 

.05741 

.06011 

.06895 

58 

59 

.04361 

.04560 

.04885 

.05136 

.05439 

.05751 

.06021 

.06407 

59 

60 

.04370 

.04569 

.04894 

.05146 

.05448 

.05762 

.06031 

.06418 

60 

251 


TABLE  XIII.— VEESINES  AND  EXSECANTS. 


/ 

20" 

21° 

22° 

23' 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

~0~ 

.06031 

.06418 

.06642 

.07115 

.07282 

.07853 

.07950 

.08636 

~T 

1 

.06041 

.06429 

.06652 

.07126 

.07293 

.07866 

.07961 

.08649 

i 

2 

.06051 

.06440 

.06663 

.07138 

.07303 

.07879 

.07972 

.08663 

2 

3 

.06061 

.06452 

.06673 

.07150 

.07314 

.07892 

.07984 

.08676 

3 

4 

.06071 

.06463 

.06684 

.07162 

.07325 

.07904 

.07995 

.08690 

4 

5 

.06081 

.06474 

.06694 

.07174 

.07336 

.07917 

.08006 

.08703 

5 

6 

.06091 

.06486 

.06705 

.07186 

.07347 

.07930 

.08018 

.08717 

6 

7 

.06101 

.06497 

.06715 

.07199 

.07358 

.07943 

.08029 

.08730 

7 

8 

.06111 

.06508 

.06726 

.07211 

.07369 

.07955 

.08041 

.08744 

8 

9 

.06121 

.06520 

.06736 

.07223 

.07380 

.07968 

.08052 

.08757 

9 

10 

.06131 

.06531 

.06747 

.07235 

.07391 

.07981 

.08064 

.08771 

10 

11 

.06141 

.06542 

.06757 

.07247 

.07402 

.07994 

.08075 

.08784 

11 

12 

.06151 

.06554 

.06768 

.07259 

.07413 

.08006 

.08086 

.08798 

12 

13 

.06161 

.06565 

.06778 

.07271 

.07424 

.08019 

.08098 

.08811 

13 

14 

.06171 

.06577 

.06789 

.07283 

.07435 

.08032 

.08109 

.08825 

14 

15 

.06181 

.06588 

.06799 

.07295 

.07446 

.08045 

.08121 

.08839 

15 

16 

.06191 

.06600 

.06810 

.07307 

.07457 

.08058 

.08132 

.08852 

16 

17 

.06201 

.06611 

.06820 

.07320 

.07468 

.08071 

.08144 

.08866 

17 

18 

.06211 

.06622 

.06831 

.07332 

.07479 

.08084 

.08155 

.08880 

18 

19 

.06221 

.06634 

.06841 

.07344 

.07490 

.08097 

.08167 

.08893 

19 

20 

.06231 

.06645 

.06852 

.07356 

.07501 

.08109 

.08178 

.08907 

20 

21 

.06241 

.06657 

.06863 

.07368 

.07512 

.08122 

.08190 

.08921 

21 

22 

.06252 

.06668 

.06873 

.07380 

.07523 

.08135 

.08201 

.08934 

22 

23 

.06262 

.06680 

.06884 

.07393 

.07534 

.08148 

.08213 

.08948 

23 

24 

.06272 

.06691 

.06894 

.07405 

.07545 

.08161 

.08225 

.08962 

24 

25 

.06282 

.06703 

.06905 

.07417 

.07556 

.08174 

.08236 

.08975 

25 

26 

.06292 

.06715 

.06916 

.07429 

.07568 

.08187 

.08248 

.08989 

26 

27 

.06302 

.06726 

.06926 

.07442 

.07579 

.08200 

.08259 

.09003 

27 

28 

.06312 

.06738 

.06937 

.07454 

.07590 

.08213 

.08271 

.09017 

23 

29 

.06323 

.06749 

.06948 

.07466 

.07601 

.08226 

.08282 

.09030 

23 

30 

.06333 

.06761 

.06958 

.07479 

.07612 

.08239 

.08294 

.09044 

30 

31 

.06343 

.06773 

.06969 

.07491 

.07623 

.08252 

.08306 

.09058 

31 

32 

.06353 

.06784 

.06980 

.07503 

.07634 

.08265 

.08317 

.09072 

32 

33 

.06363 

.06796 

.06990 

.07516 

.07645 

.08278 

.08329 

.09086 

33 

34 

.06374 

.06807 

.07001 

.07528 

.07657 

.08291 

.08340 

.09099 

34 

35 

.06384 

.06819 

.07012 

.07540 

.07668 

.08305 

.08352 

.09113 

35 

36 

.06394 

.06831 

.07022 

.07553 

.07679 

.08318 

.08364 

.09127 

36 

37 

.06404 

.06843 

.07033 

.07565 

.07690 

.08331 

.08375 

.09141 

37 

38 

.06415 

.06854 

.07044 

.07578 

.07701 

.08344 

.08387 

.09155 

38 

39 

.06425 

.06866 

.07055 

.07590 

.07713 

.08357 

.08399 

.09169 

39 

40 

.06435 

.06878 

.07065 

.07602 

.07784 

.08370 

.08410 

.09183 

40 

41 

.06445 

.06889 

.07076 

.07615 

.07735 

.08383 

.08422 

.09197 

41 

42 

.06456 

.06901 

.07087 

.07627 

.07746 

.08397 

.08434 

.09211 

42 

43 

.06466 

.06913 

.07098 

.07640 

.07757 

.08410 

.08445 

.09224 

43 

44 

.06476 

.06925 

.07108 

.07652 

.07769 

.08423 

.08457 

.09238 

44 

45 

.06486 

.06936 

.07119 

.07665 

.07780 

.08436 

.08469 

.09252 

45 

46 

.06497 

.06948 

.07130 

.07677 

.07791 

.08449 

.08481 

.09266 

46 

47 

.06507 

.06960 

.07141 

.07690 

.07802 

.08463 

.08492 

.09280 

47 

48 

.06517 

.06972 

.07151 

.07702 

.07814 

.08476 

.08504 

.09294 

48 

49 

.06528 

.06984 

.07162 

.07715 

.07825 

.08489 

.08516 

.09308 

49 

50 

.06538 

.06995 

.07173 

.07727 

.07836 

.08503 

.08528 

.09323 

50 

51 

.06548 

.07007 

.07184 

.07740 

.07848 

.08516 

.08539 

.09337 

51 

52 

.06559 

.07019 

.07195 

.07752 

.07859 

.08529 

.08551 

.09351 

52 

53 

.06569 

.07031 

.07206 

.07765 

.07870 

.08542 

.08563 

.09365 

53 

54 

.06580 

.07043 

.07216 

.0777'8 

.07881 

.08556 

.08575 

.09379 

54 

55 

.06590 

.07055 

.07227 

.07790 

.07893 

.08569 

.08586 

.09393 

55 

56 

.06600 

.07067 

.07238 

.07803 

.07904 

.08582 

.08598 

.09407 

56 

57 

.06611 

.07079 

.07249 

.07816 

.07915 

.08596 

.08610 

.09421 

57 

58 

.06621 

.07091 

.07260 

.07828 

.07927 

.08609 

.08622 

.09435 

58 

59 

.06632 

.07103 

.07271 

.07841 

.07938 

.08623 

.08634 

.09449 

59 

60 

.06642 

.07115 

.07282 

.07853 

.07950 

.08636 

.08645 

.09464 

60 

252 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


f* 

24° 

25° 

26° 

27° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.08645 

.09464 

.09309 

.10338  ; 

.10121 

.11260 

.10899 

.12233 

0 

1 

.08657 

.09478 

.09382 

.10353 

.10133 

.11276 

.10913 

.12249 

1 

2 

.08669 

.09492 

.09394 

.10368 

.10146 

.11292 

.10926 

.12266 

2 

3 

.08681 

.09506 

.09406 

.10383 

.10159 

.11308 

!  10939 

.12283 

3 

4 

.03693 

.09520 

.09418 

.10398 

.10172 

.11323 

.10952 

.12299 

4 

5 

.08705 

.09535 

.09431 

.10413 

.10184 

.11339 

.10965 

.12316 

5 

6 

.08717 

.09549 

.09443 

.10428 

.10197 

.11355 

.10979 

.12333 

6 

7 

.08728 

.09563 

.09455 

.10443 

.10210 

.11371 

.10992 

.12349 

7 

8 

.08740 

.09577 

.09468 

.10458 

.10223 

.11387 

.11005 

.12366 

8 

9 

.08752 

.09592 

.09480 

.10473 

.10236 

.11403 

.11019 

.12383 

9 

10 

.08764 

.09606 

.09493 

.10488 

.10248 

.11419 

.11032 

.12400 

10 

11 

.08776 

.09620 

.09505 

.10503 

.10261 

.11435 

.11045 

.12416 

11 

13 

.08788 

.09635 

.00517 

.10518 

.10374 

.11451 

.11058 

.12433 

12 

13 

.08800 

.09649 

.C9530 

.10533 

.10287 

.11467 

.11072 

.12450 

13 

14 

.08812 

.09663 

.09542 

.10549 

.10300 

.11483 

.11085 

.12467 

14 

15 

.08824 

.09678 

.09554 

.10564 

.10313 

.11499 

.11098 

.12484 

15 

13 

.08836 

.09692 

.09567 

.10579 

.10326 

.11515 

.11112 

.12501 

16 

17 

.08848 

.09707 

.09579 

.  10594 

.10338 

.11531 

.11125 

.12518 

17 

13 

.08860 

.09721 

.09592 

.10609 

.10351 

.11547 

.11138 

.12534 

18 

13 

.08872 

.09735 

.09604 

.10625 

.10304 

.11563 

.11152 

.12551 

19 

23 

.08884 

.09750 

.09617 

.10640 

.10377 

.11579 

.11165 

.12568 

20 

21 

.08896 

.09764 

.09629 

.10655 

.10390 

.11595 

.11178 

.12585 

21 

23 

.08903 

.09779 

.09642 

.10070 

.10403 

.11611 

.11192 

.12602 

23 

23 

.03920 

.09793 

.09654 

.10636 

.10416 

.11627 

.11205 

.12619 

23 

24 

.08932 

.09808 

.09666 

.10701 

.10429 

.11643 

.11218 

.12636 

24 

25 

.08944 

.09822 

.09679 

.10716 

.10442 

.11659 

.11232 

.12653 

25 

26 

.08956 

.09837 

.09691 

.10731 

.10455 

.11675 

.11245 

.12670 

26 

27 

.08968 

.09851 

.097'04 

.10747 

.10468 

.11691 

.11259 

.12687 

27 

28 

.08980 

.09866 

.09716 

.10762 

.10481 

.11708 

.11272 

.12704 

28 

29 

.03992 

.09880 

.09729 

.10777 

.  .10494 

.11724 

.11285 

.12721 

29 

30 

.09004 

.09895 

.09741 

.10793 

.10507 

.11740 

.11299 

.12738 

30 

31 

.09016 

.09909 

.09754 

.10808 

.10520 

.11756 

.11312 

.12755 

31 

33 

.09028 

.09924 

.09767 

.10824 

.10533 

.11772 

.11326 

.12772 

32 

33 

.09040 

.09939 

.09779 

.10839 

.10546 

.11789 

.11339 

.12789 

33 

34 

.09052 

.09953 

.09792 

.10854 

.10559 

.11805 

.11353 

.12807 

34 

35 

.09064 

.09908 

.09804 

.10870 

.10572 

.11821 

.11366 

.12824 

35 

36 

.09076 

.09982 

.09817 

.10885 

.10585 

.11838 

.11380 

.12841 

36 

37 

.09089 

.09997 

.09829 

.10901 

.10598 

.11854 

.11393 

.12858 

37 

38 

.09101 

.10012 

.09842 

.10916 

.10611 

.11870 

.11407 

.12875 

38 

39 

.09113 

.10026 

.09854 

.10932 

.10624 

.11886 

.11420 

.12892 

39 

40 

.09125 

.10041 

.09867 

.10947 

.10637 

.11903 

.11434 

.12910 

40 

41 

.09137 

.10055 

.09880 

.10963 

.10650 

.11919 

.11447 

.12927 

41 

43 

.09149 

.10071 

.09892 

.10978 

.10663 

.11936 

.11461 

.12944 

42 

43 

.09161 

.10085 

.09905 

.10994 

.10676 

.11952 

.11474 

.12961 

43 

44 

.09174 

.10100 

.09918 

.11009 

.10689 

.11968 

.11488 

.12979 

44 

45 

.09186 

.10115 

.09930 

.11025 

.10702 

.11985 

.11501 

.12996 

45 

46 

.09198 

.10130 

.09943 

.11041 

.10715 

.12001 

.11515 

.13013 

46 

47 

.09210 

.10144 

.09955 

.11056 

.10728 

.12018 

.11528 

.13031 

47 

48 

.00222 

.10159 

.09963 

.11072 

.10741 

.12034 

.11542 

.13048 

48 

49 

.09234 

.10174 

.09981 

.11087 

.10755 

.12051 

.11555 

.13065 

49 

50 

.09247 

.10189 

.09993 

.11103 

.10768 

.12067 

.11569 

.13083 

50 

51 

.09259 

.10204 

.10006 

.11119 

.10781 

.12084 

.11583 

.13100 

51 

52 

.09271 

.10218 

.10019 

.11134 

.10794 

.12100 

.11596 

.13117 

52 

53 

.09283 

.10233 

.10032 

.11150 

.10807 

.12117 

.11610 

.13135 

53 

54 

.09296 

.10248 

.10044 

.11166 

.10820 

.12133 

.11623 

.13152 

54 

55 

.09308 

.10263 

.10057 

.11181 

.10833 

.12150 

.11637 

.13170 

55 

56 

.09320 

.10278 

.10070 

.11197 

.10847 

.12166 

.11651 

.13187 

56 

57 

.09332 

.10293 

.10082 

.11213 

.10860 

.12183 

.11664 

.13205 

57 

58 

.09345 

.10308 

.10095 

.11229 

.10873 

.12199 

.11678 

.13222 

58 

59 

.09&57 

.10323 

.10108 

.11244 

.10886 

.12216 

.11692 

.13240 

59 

60 

.09369 

.10338 

.10121 

.11260 

.10899 

.12233 

.11705 

.13257 

60 

253 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


/ 

2 

8° 

2 

9" 

31 

)° 

3: 

L° 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.11705 

.13257 

.12538 

.14335 

.13397 

.15470 

.14283 

.16663 

0 

1 

.11719 

.13275 

.12552 

.14354 

.13412 

.15489 

.14298 

.16684 

1 

2 

.11733 

.13292 

.12566 

.14372 

.13427 

.15509 

.14313 

.16704 

2 

a 

.11746 

.13310 

.12580 

.14391 

.13441 

.15528 

.14328 

.16725 

3 

4 

.11760 

.13327 

.12595 

.14409 

.13456 

.15548 

.14343 

.16745 

4 

5 

.11774 

.13345 

.12609 

.14428 

.13470 

.15567 

.14358 

.16766 

5 

6 

.11787 

.13362 

.12623 

.14446 

.13485 

.15587 

.14373 

.16786 

6 

7 

.11801 

.13380 

.12637 

.14465 

.13499 

.15606 

.14388 

.16806 

•7 

8 

.11815 

.13398 

.12651 

.14483 

.13514 

.15626 

.14403 

.16827 

8 

9 

.11828 

.13415 

.12665 

.14502 

.13529 

.15645 

.14418 

.16848 

9 

10 

.11842 

.13433 

.12679 

.14521 

.13543 

.15665 

.14433 

.16868 

10 

11 

.11856 

.13451 

.12694 

.14539 

.13558 

.15684 

.14449 

.16889 

11 

12 

.11870 

.13468 

.12708 

.14558 

.13573 

.157-04 

.14464 

.16909 

12 

13 

.11883 

.13486 

.12722 

.14576 

.13587 

.15724 

.14479 

.16930 

13 

14 

.11897 

.13504 

.12736 

.14595 

.13602 

.15743 

.14494 

.1G950 

14 

15 

.11911 

.13521 

.12750 

.14014 

.13616 

.15763 

.14509 

.16971 

15 

16 

.11925 

.13539 

.12765 

.14032 

.13631 

.15782 

.14524 

.1G992 

16 

17 

.11938 

.13557 

.12779 

.14351 

.13646 

.15802 

.14539 

.17012 

17 

18 

.11952 

.13575 

.12793 

.14070 

.13660 

.15822 

.14554 

.17033 

18 

19 

.11966 

.13593 

.12807 

.14GS9 

.13G75 

.15841 

.14569 

\17C5i 

19 

20 

.11980 

.13610 

.12822 

.14707 

.13690 

.15861 

.14584 

.17075 

20 

21 

.11994 

.13628 

.12836 

.14726 

.13705 

.15881 

.14599 

.17095 

21 

22 

.12007 

.13646 

.12850 

.14745 

.13719 

.15901 

.14615 

.17116 

22 

23 

.12021 

.13664 

.12864 

.14764 

.13734 

.15920 

.14630 

.17137 

23 

24 

.12035 

.13682 

.12879 

.14782 

.13749 

.15940 

.14645 

.17158 

24 

25 

.12049 

.13700 

.12893 

.14801 

.13763 

.15960 

.146GO 

.17178 

25 

26 

.12063 

.13718 

.12207 

.14820 

.13778 

.15980 

.14675 

.17199 

26 

27 

.12077 

.13735 

.12921 

.1-1839 

.13793 

.16000 

.14690 

.17220 

27 

28 

.12091 

.13753 

.12DC6 

.14058 

.13808 

.16019 

.14706 

.17341 

23 

29 

.12104 

.13771 

.12950 

.14377 

.13822 

.10039 

.14721 

.17262 

29 

30 

.12118 

.13789 

.12964 

.14896 

.13837 

.16059 

.14736 

.17283 

30 

31 

.12132 

.13807 

.12979 

.14914 

.13852 

.16079 

.14751 

.17304 

31 

32 

.12146 

.13825 

.12993 

.14933 

.13367 

.10C99 

.147GO 

.17325 

3x5 

33 

.12160 

.13843 

.13007 

.1495.'? 

.13881 

.16119 

.14782 

.17346 

33 

34 

.12174 

.13861 

.13022 

.14971 

.13896 

.16139 

.14797 

.  7367 

34 

35 

.12188 

.13379 

.13036 

.14990 

.13911 

'  .16159 

.14812 

.  7383 

35 

36 

.12202 

.13397 

.13051 

.15009 

.13926 

.16179 

.14827 

.  7409 

36 

37 

.12216 

.13916 

.13005 

.15028 

.13941 

.16199 

.14843 

.  7430 

37 

38 

.12230 

.13934 

.13079 

.15047 

.13955 

.16219 

.14858 

.  7451 

33 

39 

.12244 

.13952 

.13094 

.15086 

.13970 

.16239 

.1487'3 

.  7472 

39 

40 

.12257 

.13970 

.13108 

.15085 

.13985 

.16259 

.14888 

.  7493 

40 

41 

.12271 

.13988 

.13122 

.15105 

.14000 

.16279 

.14904 

.  7514 

41 

42 

.12285 

.14006 

.13137 

.15124 

.14015 

.16299 

.14919 

.  7535 

42 

43 

.12299 

.14024 

.13151 

.15143 

.14030 

.16319 

.14934 

.  7556 

43 

44 

.12313 

.14042 

.131G6 

.15162 

.14044 

.16339 

.14949 

.  7577 

44 

45 

.12327 

.14061 

.13180 

.15181 

.14059 

.16359 

.149G5 

.17598 

45 

46 

.12341 

.14079 

.13195 

.15200 

.14074 

.16380 

.14980 

.17620 

46 

47 

.12355 

.14097 

.13209 

.15219 

.14089 

.16400 

.14995 

.17641 

47 

48 

.12369 

.14115 

.13223 

.15239 

.14104 

.16420 

.15011 

.17G62 

48 

49 

.12383 

.14134 

.13238 

.15258 

.14119 

.16440 

.15026 

.17G83 

49 

50 

.12397 

.14152 

.13252 

.15277 

.14134 

.16460 

.15041 

.17704 

50 

51 

.12411 

.14170 

.13267 

.15296 

.14149 

.16481 

.15057 

.17726 

51 

52 

.12425 

.14188 

.13281 

.15315 

.14164 

.16501 

.15072 

.17747 

53 

53 

.12439 

.14207 

.13296 

.15335 

.14179 

.16521 

.15087 

.17768 

53 

54 

.12454 

.14225 

.13310 

.15354 

.14194 

.16541 

.15103 

.17790 

54 

55 

.12468 

.14243 

.13325 

.15373 

.14208 

.16562 

.15118 

.17811 

55 

56 

.12482 

.14262 

.13339 

.15393 

.14223 

.16582 

.15134 

.17832 

56 

57 

.12496 

.14280 

.13354 

.15412 

.14238 

.16602 

.15149 

.17854 

57 

58 

.12510 

.14299 

.13368 

.15431 

.14253 

.16623 

.15164 

.17875 

58 

59 

.12524 

.14317 

.13383 

.15451 

.14268 

,16643 

.15180 

.17896 

59 

60 

.12538 

.14335 

.13397 

.15470 

.14283 

.16663 

.15195 

.17918 

60 

254 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


/ 

32- 

33° 

84° 

35° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.15195 

.17918 

.16133 

.19236 

.17096 

.20622 

.18085 

.22077 

0 

1 

.15211 

.17939 

.16149 

.19259 

.17113 

.20645 

.18101 

.22102 

1 

2 

.15226 

.17961 

.16165 

.19281 

.17129 

.20669 

.18118 

.22127 

2 

3 

.15241 

.17982 

.16181 

.19304 

.17145 

.20693 

.18135 

.22152 

3 

4 

.15257 

.18004 

.16196 

.19327 

.17161 

.20717 

.18152 

.22177 

4 

5 

.15272 

.18025 

.16212 

.19349 

.17178 

.20740 

.18168 

.22202 

5 

6 

.15288 

.18047 

.16228 

.19372 

.17194 

.20764 

.18185 

.22227 

6 

7 

.15303 

.18068 

.16244 

.19394 

.17210 

.20788 

.18202 

.22252 

7 

8 

.15319 

.18090 

.16200 

.19417 

.17227 

.20812 

.18218 

.22277 

8 

9 

.15334 

.18111 

.16276 

.19440 

.17243 

.20836 

.18235 

.22302 

9 

10 

.15350 

.18133 

.16292 

.19463 

.17259 

.20859 

.18252 

.22327 

10 

11 

.15305 

.18155 

.16308 

.19485 

.17276 

.20883 

.18269 

.22352 

11 

12 

.15381 

.18176 

.16324 

.19508 

.17292 

.20907 

.18286 

.22377 

12 

13 

.15396 

.18198 

.16340 

.19531 

.17308 

.20931 

.18302 

.22402 

13 

14 

.15412 

.18220 

.16355 

.19554 

.17325 

.20955 

.188*9 

.22428 

14 

15 

.15427 

.18241 

.16371 

.19576 

.17341 

.20979 

.18336 

.22453 

15 

16 

.15443 

.18263 

.16387 

.19599 

.17357 

.21003 

.18353 

.22478 

16 

17 

.15453 

.18285 

.16403 

.19622 

.17374 

.21027 

.18369 

.22503 

17 

18 

.15474 

.18307 

.16419 

.19645 

.17390 

.21051 

.18386 

.22528 

18 

19 

.15489 

.18328 

.16435 

.19668 

.17407 

.21075 

.18403 

.22554 

19 

20 

.15505 

.18350 

.16451 

.19691 

.17423 

.21099 

.18420 

.22579 

20 

21 

.15520 

.18372 

.16467 

.19713 

.17439 

.21123 

.18437 

.22604 

21 

22 

.15536 

.18394 

.16483 

.19736 

.17456 

.21147 

.18454 

.22629 

22 

23 

.15552 

.18416 

.16409 

.19759 

.17472 

.21171 

.18470 

.22655 

23 

24 

.15567 

.18437 

.16515 

.19788 

.17489 

.21195 

.18-487 

.22680 

24 

25 

.15583 

.18459 

.16531 

.19805 

.17505 

.21220 

.18504 

.22706 

25 

26 

.15598 

.18481 

.16547 

.19828 

.17522 

.21244 

.18521 

.22731 

26 

27 

.15614 

.18503 

.16563 

.19851 

.17538 

.21268 

.18538 

.22756 

27 

28 

.15630 

.18525 

.16579 

.19874 

.17554 

.21292 

.18555 

.227'82 

28 

29 

.15645 

.18547 

.16595 

.19897 

.17571 

.21316 

.18572 

.22807 

29 

30 

.15661 

.18569 

.16611 

.19920 

.17587 

.21341 

.18588 

.22833 

30 

31 

.15676 

.18591 

.16627 

.19944 

.17604 

.21365 

.18005 

.22858 

31 

32 

.15693 

.18613 

.16644 

.19967 

.17620 

.21389 

.18022 

.22884 

32 

33 

.15708 

.18635 

.16660 

.19990 

.17637 

.21414 

.18639 

.22909 

33 

34 

.15723 

.18657 

.16676 

.20013 

.17653 

.21433 

.18056 

.22935 

34 

35 

.15739 

.18679 

.16692 

.20036 

.17670 

.34402 

.18673 

.22960 

35 

36 

.15755 

.18701 

.16708 

.20059 

.17686 

.21487 

.18690 

.22986 

36 

37 

.15770 

.18723 

.16724 

.20083 

.17703 

.21511 

.18707 

.23012 

37 

38 

.15786 

.18745 

.16740 

.20106 

.17719 

.21535 

.18724 

.23037 

38 

39 

.15802 

.18767 

.16756 

.20129 

.17736 

.21500 

.18741 

.23003 

39 

40 

.15818 

.18790 

.16772 

.20152 

.17752 

.21584 

.18758 

.23089 

40 

41 

.15833 

.18812 

.16788 

.20176 

.17769 

.21609 

.18775 

.23114 

41 

42 

.15349 

.18834 

.16805 

.20193 

.17786 

.21633 

.18792 

.23140 

42 

43 

.15865 

.18856 

.16821 

.20222 

.17802 

.21658 

.18809 

.23166 

43 

44 

.15880 

.18878 

.16837 

.20246 

.17819 

.21082 

.18826 

.23192 

44 

45 

.15896 

.18901 

.16853 

.20269 

.17835 

.21707 

.18843 

.23217 

45 

46 

.15912 

.18923 

.16809 

.20292 

.17852 

.21731 

.18860 

.23243 

46 

47 

.15928 

.18945 

.16885 

.20316 

.17808 

.21756 

.18877 

.23269 

47 

48 

.15313 

.18967 

.16902 

.20339 

.17885 

.21781 

.18894 

.23295 

43 

49 

.15959 

.18990 

.16918 

.20363 

.17902 

.21805 

.18911 

.23321 

49 

50 

.15975 

.19012 

.16934 

.20386 

.17918 

.21830 

48928 

.23347 

50 

51 

.15991 

.19034 

.16950 

.20410 

.17935 

.21855 

.18945 

.23373 

51 

52 

.16006 

.19057 

.16906 

.20133 

.17952 

.21879 

.18962 

.23399 

52 

53 

.16022 

.19079 

.16983 

.20457 

.17968 

.21904 

.18979 

.23424 

53 

54 

.16038 

.19102 

.16999 

.20480 

.17985 

.21929 

.18996 

.23450 

54 

55 

.16054 

.19124 

.17015 

.20504 

.18001 

.21953 

.19013 

.23476 

55 

56 

.16070 

.19146 

.17031 

.20527 

.18018 

.21978 

.19030 

.23502 

56 

57 

.16085 

.19169 

.17047 

.20551 

.18035 

.22003 

.19047 

.23529 

57 

58 

.16101 

.19191 

.17064 

.20575 

.18051 

.22028 

.19064 

.23555 

58 

59 

.16117 

.19214  ; 

.17080 

.20598 

.18068 

.22053 

.19081 

.23581 

59 

60 

.16133 

.19236  I 

.17096 

.20622 

.18085 

.22077  1 

.19098 

.23607 

60 

255 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


t 

36° 

37° 

38° 

39« 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.19008 

.23007 

.201S6 

.25214 

.21199 

.20902 

.22285 

.28076 

~0~ 

1 

.19115 

.23633 

.20154 

.25241 

.21217 

.26931 

.22304 

.28706 

1 

2 

.19133 

.23659 

.20171 

.25269 

.21235 

.20960 

.22322 

.28737 

2 

3 

.19150 

.23685 

.20189 

.25296 

.21253 

.20988 

.22340 

.28767 

3 

4 

.19167 

.23711 

.20207 

.25324 

.21271 

.27017 

.22359 

.28797 

4 

5 

.19184 

.23738 

.20224 

.25351 

.21289 

.27046 

.22377 

.28823 

5 

6 

.19201 

.23764 

.20242 

.25379 

.21307 

.27075 

.22395 

.28858 

6 

7 

.19218 

.23790 

.20259 

.25406 

.21324 

.27104 

.22414 

.28889 

7 

8 

.19235 

.23816 

.20277 

.25434 

.21342 

.27133 

.22432 

.28919 

8 

9 

.19252 

.23843 

.20294 

.25402 

.21300 

.27162 

.22450 

.28950 

9 

10 

.19270 

.23869 

.20312 

.25489 

.21378 

.27191 

.22469 

.28980 

10 

11 

.19287 

.23895 

.20329 

.25517 

.21396 

.27221 

.22487 

.29011 

11 

12 

.19304 

.23922 

.20347 

.23545 

.21414 

.27250 

.22506 

.29042 

12 

13 

.19321 

.23948 

.20305 

.25572 

.21432 

.27279 

.22524 

.29072 

13 

14 

.19338 

.23975 

.20382 

.25600 

.21450 

.27308 

.22542 

.29103 

14 

15 

.19356 

.24001 

.20400 

.25628 

.21468 

.27337 

.22561 

.29133  1  15 

16 

.19873 

.24028 

.20417 

.25656 

.21486 

.27366 

.22579 

.29164  16 

17 

.19390 

.24054 

.20435 

.25683 

.21504 

.27396 

.22598 

.29195 

17 

18 

.19407 

.24081 

.20453 

.25711 

.21522 

.27425 

.22616 

.29226 

18 

19 

.19424 

.24107 

.20470 

.25739 

.21540 

.27454 

.22634 

.29256 

19 

20 

.19442 

.24134 

.20488 

.25767 

.21558 

.27483 

.22653 

.29287 

20 

21 

.19459 

.24160 

.20506 

.25795 

.21576 

.27513 

.22671 

.29318 

21 

22 

.19476 

.24187 

.20523 

.25823 

.21595 

.27542 

.22090 

.29349 

23 

23 

.19493 

.24213 

.20541 

.25851 

.21613 

.27572 

.22703 

.29380 

23 

24 

.19511 

.2-1240 

.20559 

.25879 

.21631 

.27601 

.22727 

.29411 

24 

25 

.19528 

.2-4267 

.20576 

.25907 

.21649 

.27630 

.22745 

.29442 

25 

26 

.19545 

.24293 

.20594 

.25935 

.21667 

.27660 

.22704 

.29473 

26 

27 

.19562 

.24320 

.20612 

.25963 

.21685 

.27689 

.22782 

.29504 

27 

28 

.19580 

.24347 

.20029 

.25991 

.21703 

.27719 

.22801 

.29535 

28 

29 

.19597 

.24373 

.20647 

.26019 

.21721 

.27748 

.22819 

.29566 

29 

30 

.19614 

.24400 

.20665 

.26047 

.21739 

.27775 

.22838 

.29597 

30 

31 

.19632 

.24427 

.20682 

.26075 

.21757 

.27807 

.22856 

.29628 

31 

32 

.19049 

.24454 

.20700 

.20104 

.21775 

.27837 

.22875 

.29659 

32 

33 

.19666 

.24481 

.20718 

.26132 

.21794 

.27867 

.22893 

.29090 

33 

34 

.Iy684 

.24508 

.20736 

.26160 

.21812 

.27896 

.22912 

.29721 

34 

35 

.19701 

.24534 

.20753 

.26188 

.21830 

.27926 

.22330 

.29752 

35 

30 

.19718 

.24561 

.20771 

.26216 

.21848 

.27956 

.22949 

.29784 

SG 

37 

.19736 

.24588 

.20789 

.26245 

.21866 

.27985 

.22907 

.29815 

37 

38 

.19753 

.24615 

.20807 

.26273 

.21884 

.28015 

.22986 

.29846 

33 

39 

.19770 

.21642 

.20824 

.26301 

.21902 

.28045 

.23004 

.29877 

39 

40 

.19788 

.24669 

.20842 

.26330 

.21921 

.28075 

.23023 

.29909 

40 

41 

.19805 

.24696 

.20860 

.26358 

.21939 

.28105 

.23041 

.29940 

41 

42 

.19822 

.24723 

.20878 

.26387 

.21957 

.28134 

.23000 

.29971 

42 

43 

.19840 

.24750 

.20895 

.26415 

.21975 

.28104 

.23079 

.30003 

43 

44 

.19857 

.24777 

.20913 

.26443 

.21993 

.28194 

.23097 

.30034 

44 

45 

.1987'5 

.24804 

.20931 

.26472 

.22012 

.28224 

.23116 

.30006 

45 

46 

.19892 

.24832 

.20949 

.26500 

.22030 

.28254 

.23134 

.30097 

46 

47 

.19909 

.24859 

.20967 

.26529 

.22048 

.28284 

.23153 

.30129 

47 

48 

.19927 

.24886 

.20985 

.26557 

.22006 

.28314 

.28178 

.30100 

48 

49 

.19944 

.24913 

.21002 

.26586 

.22084 

.28344 

.23190 

.30192 

49 

50 

.19962 

.24940 

.21020 

.26615 

.22103 

.28374 

.23209 

.30223 

50 

51 

.19979 

.24967 

.21038 

.26643 

.22121 

.28404 

.23228 

.30255 

51 

52 

.19997 

.24995 

.21056 

.26672 

.22139 

.28434 

.23246 

.30287 

52 

53 

.20014 

.25022 

.21074 

.20701 

.22157 

.28464 

.23265 

.30318 

53 

54 

.20032 

.25049 

.21092 

.20729 

.22176 

.28495 

.23283 

.30350 

54 

55 

.20049 

.25077 

.21109 

.26758 

.22194 

.28525 

.23302 

.30382 

55 

56 

.20066 

.25104 

.21127 

.26787 

.22212 

.28555 

.23321 

.30413 

56 

57 

.20084 

.25131 

.21145 

.26815 

.22231 

.28585 

.23339 

.30445 

57 

58 

.20101 

.25159 

.21163 

.26844 

.22249 

.28615 

.23358 

.30477 

58 

59 

.20119 

.25186 

.21181 

.26873 

.22267 

.28646 

.23377 

.30509 

59 

60 

.20136 

.25214  1 

.21199 

.26902 

.22285 

.28676 

.23396 

.30541 

60 

256 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


1 

40° 

41° 

42« 

43° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.23396 

.30541 

.24529 

.32501 

.25686 

.34563 

.26865 

.36733 

"o" 

1 

.23414 

.30573 

.24548 

.32535 

.25705 

.3-1599 

.26884 

.36770 

1 

2 

.23433 

.30605 

.24567 

.32568 

.25724 

.34634 

.26904 

.36807 

2 

3 

.23452 

.30636 

.24586 

.32602 

.25744 

.34669 

.26924 

.36844 

8 

4 

.23470 

.30668 

.24605 

.32636 

.25763 

.34704 

.26944 

.36881 

4 

5 

.23489 

.30700 

.24625 

.32669 

.25783 

.34740 

.26964 

.36919 

6 

6 

.23508 

.30732 

.24644 

.32703 

.25802 

.34775 

.26984 

.36956 

6 

7 

.23527 

.30764 

.24663 

.32737 

.25822 

.34811 

.27004 

.36993 

7 

8 

.23545 

.30796 

.24682 

.32770 

.25841 

.34846 

.27024 

.37030 

8 

9 

.23564 

.30829 

.24701 

32804 

.25861 

.34882 

.27043 

.37068 

9 

10 

.23583 

.30861 

.24720 

.32838 

.25880 

.34917 

.27063 

.37105 

10 

11 

.23602 

.30893 

.24739 

.32872 

.25900 

.34953 

.27083 

.37143 

11 

12 

.23620 

.30925 

.24759 

.32905 

.25920 

.34988 

.27103 

.37180 

12 

13 

.23639 

.30957 

24778 

.32939 

.25939 

.35024 

.27123 

.37218 

13 

14 

.23658 

.30989 

24797 

.32973 

.25959 

.35060 

.27143 

.37255 

14 

15 

.23677 

.31022 

.24816 

.33007 

.25978 

.35095 

.27163 

.37223 

15 

16 

.23696 

.31054 

.24835 

.33041 

.25998 

.35131 

.27183 

.37330 

16 

17 

.23714 

.31086 

.24854 

.33075 

.26017 

.35167 

.27203 

.37368 

17 

18 

.23733 

.31119 

.24874 

.33109 

.26037 

.35203 

.27223 

.37406 

18 

19 

.23752 

.31151 

.24893 

.33143 

.26056 

.35238 

.27243 

.37443 

19 

20 

.23771 

.31183 

.24912 

.33177 

.26076 

.35274 

.27263 

.37481 

20. 

21 

.23790 

.31216 

.24931 

.33211 

.26096 

.35310 

.27283 

.37519 

21 

22 

.23808 

.31248 

.24950 

.33245 

.26115 

.35346 

.27303 

.37556 

22 

23 

.23827 

.31281 

.24970 

.33279 

.26135 

.35382 

.27323 

.37594 

23 

24 

.23846 

.31313 

.24989 

.33314 

.26154 

.35418 

.27343 

.37632 

24 

25 

.23865 

.31346 

.25008 

.33348 

.26174 

.35454 

.27363 

.37670 

25 

26 

.23884 

.31378 

.25027 

.33382 

.26194 

.35490 

.27383 

.37708 

26 

27 

.23903 

.31411 

.25047 

.33416 

.26213 

.35526 

.27403 

.37746 

27 

28 

.23922 

.31443 

.25066 

.33451 

.26233 

.35562 

.27423 

.37784 

28 

29 

.23941 

.31476 

.25085 

.33485 

.26253 

.35598 

.27443 

.37822 

29 

30 

...23959 

.31509 

.25104 

.33519 

.26272 

.35634 

.27463 

.87860 

30 

31 

;23978 

.31541 

.25124 

.33554 

.26292 

.35670 

.27483 

.87898 

31 

32 

.23997 

.31574 

.25143 

.33588 

.26312 

.35707 

.27503 

.37936 

32 

33 

.24016 

.31607 

.25162 

.33622 

.26331 

.35743 

.27523 

.87974 

33 

34 

.24035 

.31610 

.25182 

.33657 

.26351 

.35779 

.27543 

.88012 

34 

35 

.24054 

.31672 

.25201 

.33691 

.26371 

.35815 

.27563 

.38051 

35 

36 

.24073 

.31705 

.25220 

.33726 

.26390 

.35852 

.27583 

.38089 

36 

37 

.24092 

.31738 

.25240 

.33760 

.26410 

.35888 

.27603 

.88127 

37 

38 

.24111 

.31771 

.25259 

.33795 

.26430 

.35924 

.27623 

.38165 

38 

39 

.24130 

.31804 

.25278 

.33830 

.26449 

.35961 

.27643 

.38204 

39 

40 

.24149 

.31837 

.25297 

.33864 

.26469 

.35997 

.27663 

.38242 

40 

41 

.24168 

.31870 

.25317 

.33899 

.26489 

.36034 

.27683 

.38280 

41 

42 

.24187 

.31903 

.25336 

.33934 

.26509 

.36070 

.27703 

.38319 

42 

43 

.24206 

.31936 

.25356 

.33968 

.26528 

.36107 

.27723 

.38357 

43 

44 

.24225 

.31969 

.25375 

.34003 

.26548 

.36143 

.27743 

.38396 

44 

45 

.24244 

.32002 

.25394 

.34038 

.26568 

.36180 

.27764 

.38431 

45 

46 

.24262 

.32035 

.25414 

.34073 

.26588 

.36217 

.27784 

.38473 

46 

47 

.24281 

.32068 

.25433 

.34108 

.26607 

.36253 

.27804 

.38512 

47 

48 

.24300 

.32101 

.25452 

.34142 

.26627 

.36290 

.27824 

.38550 

48 

49 

.24320 

.32134 

.25472 

.34177 

.26647 

.36327 

.27844 

.38589 

49 

50 

.24339 

.32168 

.25491 

.34212 

.26667 

.36363 

.27864 

.38628 

50 

51 

.24358 

.32201 

.25511 

.34247 

.26686 

.36400 

.27884 

.38666 

51 

52 

.24377 

.32234 

.25530 

.34282 

.26706 

.36437 

.27905 

.38705 

52 

53 

.24396 

.32267 

.25549 

.34317 

.26726 

.36474 

.27925 

.38744 

53 

54 

.24415 

.32301 

.25569 

.34352 

.26746 

.36511 

.27945 

.38763 

54 

55 

.24434 

.32334 

.25588 

.34387 

.26766 

.36548 

.27965 

.38822 

55 

56 

.24453 

.32368 

.25608 

.34423 

.26785 

.36585 

.27985 

.38860 

56 

57 

-.24472 

.32401 

.25627 

.34458 

.26805 

.36622 

.28005 

.38899 

&: 

58 

.24491 

.32434 

.25647 

.84493 

.26825 

.36G59 

.28026 

.38938 

58 

59 

.24510 

,32468 

,25666 

.34528 

.26845 

.36696 

.28046 

.38977 

59 

60 

.24529 

.32501 

.25686 

.34563 

.26865 

.36733 

.28066 

.39016 

60 

TABLE  XIII.— VERSINES  AND  EXSECANTS. 


' 

44° 

45° 

46° 

47° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec.  , 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.28066 

.39016 

.29289 

.41421 

.305.3-4 

.43956 

.31800 

.46628 

0 

1 

.28086 

.39055 

.29310 

.41463 

.30555 

.43999 

.31821 

.46674 

1 

2 

.28106 

.39095 

.29330 

.41504 

.30576 

.44042 

.31843 

.46719 

2 

3 

.28127 

.39134 

.29351 

.41545 

.30597 

.44086 

.31864 

.46765 

3 

4 

.28147 

.39173 

.29372 

.41586 

.30618 

.44129 

.81885 

.46811 

4 

5 

.28167 

.39212 

.29392 

.41627 

.30639 

.44173 

.31907 

.46857 

5 

6 

.28187 

.39251 

.29413 

.41669 

.30660 

.44217 

.31928 

.46903 

6 

7 

.28208 

.39291 

.29433 

.41710 

.30081 

.44260 

.31949 

.46949 

7 

8 

.28228 

.39330 

.29454 

.41752 

.30702 

.44304 

.31971 

.46995 

8 

9 

.28248 

.39369 

.29475 

.41793 

.30723 

.44347 

.31992 

.47041 

9 

10 

.28268 

.39409 

.29495 

.41835 

.30744 

.44391 

.32013 

.47087 

10 

11 

.28289 

.39448 

.29516 

.41876 

.30765 

.44435 

.32035 

.47134 

11 

12 

.28309 

.39487 

.29537 

.41918 

.30786 

.44479 

.32056 

.47180 

12 

13 

.28329 

.39527 

.29557 

.41959 

.30807 

.44523 

.32077 

.47226 

13 

14 

.28350 

.39566 

.29578 

.42001 

.30828 

.44567 

.32099 

.47272 

14 

15 

.28370 

.39606 

.29599 

.42042 

.30849 

.44610 

.32120 

.47319 

15 

16 

.28390 

.39646 

.29619 

.42084 

.30870 

.44654 

.32141 

.47365 

16 

17 

.28410 

.39685 

.29640 

.42126 

.30891 

.44698 

.32163 

.47411 

17 

18 

.28431 

.39725 

.29661 

.42168 

.30912 

.44742 

.32184 

.47458 

18 

19 

.28451 

.39764 

.29G81 

.42210 

.30933 

.44787 

.32205 

.47501 

19 

20 

.28471 

.39804 

.29702 

.42251 

.30954 

.44831 

.32227 

.47551 

20 

21 

.28492 

.39844 

.29723 

.42293 

.30975 

.44875 

.32248 

.47598 

21 

22 

.28512 

.39384 

.29743 

.42335 

.30933 

.44919 

.32270 

.47644 

22 

23 

.28532 

.39924 

.2970  i 

.42377 

.31017 

.44963 

.32291 

.47691 

23 

24 

.28553 

.39963 

.29785 

.42419 

.31038 

.45007 

.32312 

.  47738 

24 

25 

.28573 

.40003 

.29805 

.42461 

.31059 

.45052 

.32334 

.47784 

25 

26 

.28503 

.40043 

.29826 

.42503 

.31030 

.45096 

.323.')5 

.47831 

26 

27 

.28614 

.40083 

.29847 

.42545 

.31101 

.45141 

.32377 

.47878 

27 

28 

.28634 

.40123 

.29868 

.42587 

.31122 

.45185 

.32398- 

.47925 

28 

29 

.28655 

.40163 

.29888 

.42630 

.31143 

.45229 

.32420 

.47972 

29 

30 

.28675 

.40203 

.29909 

.42672 

.31165 

.45274 

.32441 

.48019 

30 

31 

.28695 

.40243 

.29930 

.42714 

.31186 

.45319 

.32462 

.48066 

31 

32 

.28716 

.40283 

.29951 

.42756 

.31207 

.45363 

.32-184 

.48113 

Si 

33 

.28736 

.40324 

.29971 

.42799 

.31228 

.45408 

.32505 

.48160 

33 

34 

.28757 

.40304 

.29992 

.42841 

.31219 

.45452 

.32527 

.48207 

34 

35 

.28777 

.40104 

.30013 

.42883 

.81270 

.45497 

.32548 

.48254 

35 

36 

.28797 

.40414 

.30034 

.42926 

.31291 

.45542 

.32570 

.48301 

36 

37 

.28818 

.40485 

.30054 

.42908 

.31312 

.45587 

.32591 

.48349 

37 

38 

.28838 

.40525 

.30075 

.43011 

.31334 

.45031 

.32613 

.48396 

38 

39 

.28859 

.40r,G5 

.30096 

.43053 

.31355 

.45676 

.32634 

.48443 

39 

40 

.28879 

.40606 

.30117 

.43096 

.31376 

.45721 

.32656 

.48491 

40 

41 

.28900 

.40646 

.30138 

.43139 

.31397 

.45766 

.32677 

.48538 

41 

42 

.28920 

.40687 

.30158 

.43181 

.31418 

.45811 

.32699 

.48586 

42 

43 

.28941 

.40727 

.30179 

43224 

.31439 

.45856 

.32720 

.48633 

43 

44 

.28961 

.40768 

.30200 

.43267 

.31461 

.45901 

.32742 

.48681 

44 

45 

.28981 

.40808 

.30221 

.43310 

.31482 

.45946 

.32763 

.48728 

45 

46 

.29002 

.40849 

.30242 

.43352 

.31503 

.45992 

.32785 

.48776 

46 

47 

.29022 

.40890 

.30263 

.433D5 

.31524 

.46037 

.32806 

.48824 

47 

48 

.29043 

.40930 

.30283 

.43438 

.31545 

.46082 

.32828 

.48871 

48 

49 

.29063 

.40971 

.30304 

.43481 

.31567 

.46127 

.32849 

.48919 

49 

50 

.29084 

.41012 

.30325 

.43524 

.31588 

.46173 

.32871 

.48967 

50 

51 

.29104 

.41053 

.30346 

.43567 

.31609 

.46218 

.32893 

.49015 

51 

52 

.29125 

.41093 

.30367 

.43610 

.31630 

.46203 

.32914 

.49063 

52 

53 

.29145 

.41134 

.30388 

.43653 

.31651 

.46309 

.32936 

.49111 

53 

54 

.29166 

.41175 

.30409 

.43696 

.31073 

.46354 

.32957 

.49159 

54 

55 

.29187 

.41216 

.30430 

.43739 

.31694 

.46400 

.32979 

.49207 

55 

56 

.29207 

.41257 

.30451 

.43783 

.31715 

.46445 

.33001 

.49255 

56 

57 

.29228 

.41298 

.30471 

.43826 

.31736 

.46491 

.33022 

.49303 

57- 

58 

.29248 

.41339 

.30492 

.43869 

.31758 

.46537 

.33044 

.49351 

58 

59 

.29269 

.41380 

.30513 

.43912 

.31779 

.46582 

.33065 

.49399 

59 

60 

.29289 

.41421 

.30534 

.43956 

.31800 

.46628 

.33087 

.49448 

60 

•258 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


48- 

49s 

50° 

51° 

r 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

o 

.33087 

.49448 

.34394 

.52425 

.35721 

.55572 

.37068 

.58902 

0 

1 

.33109 

.49496 

.34416 

.52476 

.35744 

.55626 

.37091 

.58959 

1 

2 

.33130 

.49544 

.34438 

.52527 

.35766 

.55680 

.37113 

.59018 

2 

3 

.33152 

.49593 

.31460 

.52579 

.35788 

.55734 

.37136 

.59073 

3 

4 

33173 

.49641 

.34482 

.52630 

.35810 

.55789 

.37158 

.59130 

4 

5 

.33195 

.49690 

.34504 

.52681 

.35833 

.55843 

.37181 

.59188 

5 

6 

.33217 

.49738 

.34526 

.52732 

.35855 

.55897 

.37204 

.59245 

6 

7 

.33238 

.49787 

.34548 

.52784 

.35877 

.55951 

.37226 

.59302 

7 

8 

.33260 

.49835 

.34570 

.52835 

.35900 

.56005 

.37249 

.59360 

8 

9 

.33282 

.49884 

.34592 

.52886 

.35922 

.56060 

.37272 

.59418 

9 

10 

.33303 

.49933 

.34614 

.52938 

.35944 

.56114 

.37294 

.59475 

10 

11 

.33325 

.49981 

.34636 

.52989 

.35967 

.56169 

.37317 

.59533 

11 

12 

.33347 

.50030 

.34638 

.53041 

.35989 

.56223 

.37340 

.59590 

12 

13 

.33368 

.50079 

.34680 

.53C92 

.36011 

.56278 

.37362 

.59648 

13 

14 

.33390 

.50128 

.34702 

.53144 

.36034 

.56332 

.37385 

.59706 

14 

15 

.33412 

.50177 

.34724 

.53196 

.36056 

.56387 

.37408 

.59764 

15 

16 

.33434 

.50226 

.34746 

.53247 

.36078 

.56442 

.37430 

.59822 

16 

17 

.33455 

.50275 

.34768 

.53299 

.36101 

.56497 

.37453 

.59880 

17 

18 

.33477 

.50324 

.34790 

.53351 

.36123 

.56551 

.37476 

.59938 

13 

19 

.33499 

.50373 

.34812 

.53403 

.36146 

.56606 

.37493 

.59996 

19 

20 

.33520 

.50422 

.34834 

.53455 

.36168 

.56661 

.37521 

.60054 

20 

21 

.33542 

.50471 

.34856 

.53507 

.36190 

.56716 

.37544 

.60112 

21 

22 

.33564 

.50521 

.34878 

.53559 

.36213 

.56771 

.37567 

.60171 

23 

23 

.33336 

.50570 

.34900 

.53611 

.36235 

.56826 

.37589 

.60229 

23 

24 

.33607 

.50619 

.34923 

.53663 

.36258 

.56881 

.37612 

.60287 

24 

25 

.33629 

.50669 

.34945 

.53715 

.36230 

.56937 

.37635 

.60346 

25 

26 

.33651 

.50718 

.34967 

.53768 

.36302 

.56992 

.37658 

.60404 

26 

27 

.33673 

.50767 

.34989 

.53820 

.36325 

.57047 

.37680 

.60463 

27 

28 

.33694 

.50817 

.35011 

.53872 

.36347 

.57103 

.37703 

.60521 

28 

29 

.33716 

.50866 

.350S3 

.53924 

.36370 

.57158 

.37726 

.60580 

29 

30 

^33738 

.50916 

.35055 

.53977 

.36392 

.57213 

.37749 

.60639 

30 

31 

-.183760 

.50966 

.35077 

.54029 

.36415 

.57269 

.37771 

.60698 

81 

32 

.33782 

.51015 

.35099 

.54082 

.36437 

.57324 

.37794 

.607'36 

32 

33 

.33803 

.51065 

.35122 

.54134 

.36460 

.57380 

.37817 

.60815 

2? 

34 

.33825 

.51115 

.35144 

.54187 

.36482 

.57436 

.37840 

.60874 

34 

35 

.33847 

.51165 

.35166 

.54240 

.36504 

.57491 

.37862 

.60033 

35 

36 

.33869 

.51215 

.35188 

.54292 

.36527 

.57547 

.37885 

.60992 

S3 

37 

.33891 

.51265 

.35210 

.54345 

.36549 

.57603 

.37908 

.61051 

37 

38 

.33912 

.51314 

.352S2 

.54398 

.36572 

.57659 

.37931 

.61111 

33 

39 

.33934 

.51364 

.35254 

.54451 

.36594 

.57715 

.37954 

.61170 

39 

40 

.33956 

.51415 

.35277 

.54504 

.36617 

.57771 

.37976 

.61229 

40 

41 

.a3978 

.51465 

.35299 

.54557 

.36639 

.57827 

.37999 

.61288 

41 

42 

.34000 

.51515 

.35321 

.54610 

.36662 

.57883 

.38022 

.61348 

42 

43 

.34022 

.51565 

.35343 

.54063 

.36684 

.57939 

.38045 

.61407 

43 

44 

.34044 

.51615 

.35365 

.54716 

.36707 

.57995 

.38068 

.61467 

44 

45 

.34065 

.51665 

.35388 

.54769 

.36729 

.58051 

.38091 

.61526 

45 

46 

.34087 

.51716 

.35410 

.54822 

.36752 

.58108 

.38113 

.61586 

46 

47 

.34109 

.51766 

.35432 

.54876 

.36775 

.58164 

.38136 

.61646 

47 

48 

.34131 

.51817 

.35454 

.54929 

.36797 

.58221 

.38159 

.61705 

48 

49 

.34153 

.51867 

.35476 

.54982 

.36820 

.58277 

.38182 

.61765 

49 

50 

.34175 

.51918 

.35499 

.55036 

.36842 

.58333 

.38205 

.61825 

50 

51 

.34197 

.51968 

.35521 

.55089 

.36865 

.58390 

.38228 

.61885 

51 

52 

.34219 

.52019 

.35543 

.55143 

.36887 

.58447 

.38251 

.61945 

52 

53 

.34241 

.52069 

.35565 

.55196 

.36910 

.58303 

.38274 

.62005 

53 

54 

.34262 

.52120 

.35388 

.55250 

.36932 

.58560 

.38296 

.62065 

54 

55 

.34284 

.52171 

.35610 

.55303 

.36955 

.58017 

.38319 

.62125 

55 

56 

.34306 

.52222 

.35632 

.55357 

.36978 

.58674 

.38342 

.62185 

56 

57 

A?34328 

.52273 

.35654 

.55411 

.37000 

.58731 

.38365 

.62246 

57;- 

58 

.'34350 

.52323 

.35677 

.55465 

.37023 

.58788 

.38388 

.62306 

58' 

59 

.34372 

.52374 

.35699 

.55518 

.37045 

.58845 

.38411 

.62366 

59 

60 

.84394 

.52425 

.35721 

.55572 

.37068 

.58902 

.38434 

.62427 

60 

259 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


/ 

52° 

53° 

54° 

55° 

/ 

Vers. 

Exsec. 

Vers. 

/ 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.38434 

.62427 

.39819 

.66164 

.41221 

.70130 

.42642 

.74345 

0 

1 

.38457 

.62487 

.39842 

.66228 

.41245 

.70198 

.42666 

.74417 

1 

2 

.38480 

.62548 

.39865 

.66292 

.41269 

.70267 

.42690 

.74490 

2 

3 

.38503 

.62609 

.39888 

.66357 

.41292 

.70335 

.42714 

.74562 

3 

4 

.38526 

.62669 

.39911 

.66421 

.41316 

.70403 

.42738 

.74635 

4 

5 

.38549 

.62730 

.39935 

.66486 

.41339 

.70472 

.42762 

.74708 

5 

6 

.38571 

.62791 

.39958 

.66550 

.41363 

.70540 

.42785 

.74781 

6 

7 

.38594 

.62852 

.39981 

.66615 

.41386 

.70609 

.42809 

.74854 

7 

8 

.38617 

.62913 

.40005 

.66679 

.41410 

.70677 

.42833 

.74927 

8 

9 

.38640 

.62974 

.40028 

.66744 

.41433 

.70746 

.42857 

.75000 

9 

10 

.38663 

.63035 

.40051 

.66809 

.41457 

.70815 

.42881 

.75073 

10 

11 

.38686 

.63096 

.40074 

.66873 

.41481 

.70884 

.42905 

.75146 

11 

12 

.38709 

.63157 

.40098 

.66938 

.41504 

.70953 

.42929 

.75219 

12 

13 

.38732 

.63218 

.40121 

.67003 

.41528 

.71022 

.42953 

.75293 

13 

14 

.38755 

.63279 

.40144 

.67068 

.41551 

.71091 

.42976 

.75366 

14 

15 

.38778 

.63341 

.40168 

.67133 

.41575 

.71160 

.43000 

.75440 

15 

16 

.38801 

.63402 

.40191 

.67199 

.41599 

.71229 

.43024 

.75513 

16 

17 

.38824 

.63464 

.40214 

.67264 

.41622 

.71298 

.43048 

.75587 

17 

18 

.38847 

.63525 

.40237 

.67329 

.41646 

.71368 

.43072 

.75661 

18 

19 

.38870 

.63587 

.4026? 

.67394 

.41670 

.71437 

.43096 

.75734 

19 

20 

.38893 

.63648 

.40284 

:  67460 

.41693 

.71506 

.43120 

.75808 

20 

21 

.38916 

.63710 

.40307 

.67525 

.41717 

.71576 

.43144 

.75882 

21 

22 

.38939 

.63772 

.40331 

.67591 

.41740 

.71646 

.43168 

.75956 

22 

23 

.38962 

.63834 

.40354 

.67656 

.41764 

.71715 

.43192 

.76031 

23 

24 

.38985 

.63895 

.40378 

.67722 

.41788 

.71785 

.43216 

.76105 

24 

25 

.39009 

.63957 

.40401 

.67788 

.41811 

.71855 

.43240 

.76179 

25 

26 

.39032 

.64019 

.40424 

.67853 

.41835 

.71925 

.43264 

.76253 

26 

27 

.39055 

.64081 

.40448 

.67919 

.41859 

.71995 

.43287 

.76328 

27 

28 

.39078 

.64144 

.40471 

.67985 

41882 

.72065 

.43311 

.76402 

28 

29 

.39101 

.64206 

.40494 

.68051 

41906 

.72135 

.43335 

.76477 

29 

30 

.39124 

.64268 

.40518 

.68117 

.41930 

.72205 

.43359 

.76552 

30 

31 

.39147 

.64330 

.40541 

.68183 

.41953 

.72275 

.43383 

.76626 

31 

32 

.39170 

.64393 

.40565 

68250 

.41977 

.72346 

.43407 

.76701 

32 

33 

.39193 

.64455 

.40588 

.68316 

.42001 

.72416 

.43431 

.76776 

33 

34 

.39216 

.64518 

.40011 

68382 

.42024 

.72487 

.43455 

.76851 

34 

85 

.39239 

.64580 

.40635 

!  68449 

.42048 

.72557 

.43479 

.76926 

35 

36 

.39262 

.64643 

.40658 

.68515 

.42072 

.72628 

.43503 

.77001 

36 

37 

.39286 

.64705 

.40682 

.68582 

.42096 

.72698 

.43527 

.77077 

37 

38 

.39309 

.04768 

.40705 

.68648 

.42119 

.72769 

.43551 

.77152 

38 

39 

.39332 

.64831 

.40728 

.68715 

.42143 

.72840 

.43575 

.77227 

39 

40 

.39355 

.64894 

.40752 

.68782 

.42167 

.72911 

.43599 

.77303 

40 

41 

.39378 

.64957 

.40775 

.68848 

.42191 

.72982 

.43623 

.77378 

41 

42 

.39401 

.65020 

.40799 

.68915 

.42214 

.73053 

.43647 

.77454 

42 

43 

.39424 

.65083 

.40822 

.68982 

.42238 

.73124 

.43671 

.77530 

43 

44 

.39447 

.65146 

.40846 

.69049 

.42262 

.73195 

.43695 

.77606 

44 

45 

.39471 

.65209 

.40869 

.69116 

.42285 

.73267 

.43720 

.77681 

45 

46 

.39494 

.65272 

.40893 

.69183 

.42309 

.73338 

.43744 

.77757 

46 

47 

.39517 

.65336 

.40916 

.69250 

.42333 

.73409 

.43768 

.77833 

47 

48 

.39540 

.65399 

.40939 

.69318 

.42357 

.73481 

.43792 

.77910 

48 

49 

.39563 

.65462 

.409G3 

.69385 

.42381 

.73552 

.43816 

.77986 

49 

50 

.39586 

.65526 

.40986 

.69452 

.42404 

.73024 

.43840 

.78062 

50 

51 

.39610 

.65589 

.41010 

.69520 

.42428 

.73696 

.43864 

.78138 

51 

52 

.39633 

.65653 

.41033 

.69587 

.42452 

.73768 

.43888 

.78215 

62 

53 

.39656 

.65717 

.41057 

.69655 

.42476 

.73840 

.43912 

.78291 

53 

54 

.39679 

.65780 

.41080 

.69723 

.42499 

.73911 

.43936 

.7'8368 

54 

55 

.39702 

.65844 

.41104 

.69790 

.42523 

.73983 

.43960 

.78445 

55 

56 

.39726 

.65908 

.41127 

.69858 

.42547 

74056 

.43984 

.78521 

56 

57 

.39749 

.65972 

.41151 

.69926 

.42571 

.74128 

.44008 

.78598 

57  , 

58 

.39772 

.66036 

.41174 

.69994 

.42595 

.74200 

.44032 

.78675 

58 

59 

.39795 

.66100 

.41198 

.70062 

.42619 

.7427'2 

.44057 

.78752 

59 

60 

.39819 

.66164 

.41221 

.70130 

.42642 

.74345 

.44081 

.78829 

60 

TABLE  XIII.— VERSINES  AND  EXSECANTS. 


/ 

56' 

57° 

58- 

59° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.44081 

.78829 

.45536 

.83608 

.47008 

.88708 

.48496 

.94160 

0 

1 

.44105 

.78906 

.45560 

.83690 

.47033 

.88796 

.48521 

.94254 

1 

2 

.44129 

.78984 

.45585 

.83773 

.47057 

.88884 

.48546 

.94349 

2 

3 

.44153 

.79061 

.45609 

.83855 

.47082 

.88972 

.48571 

.94443 

3 

4 

.44177 

.79138 

.45634 

.83938 

.47107 

.89060 

.48596 

.94537 

4 

5 

.44201 

.79216 

.45658 

.84020 

.47131 

.89148 

.48621 

.94632 

5 

6 

.44225 

.79293 

.45683 

.84103 

.47156 

.89237 

.48646 

.94726 

6 

7 

.44250 

.79371 

.45707 

.84186 

.47181 

.89325 

.48671 

.94821 

7 

8 

.44274 

.79449 

.45731 

.84269 

.47206 

.89414 

.48696 

.94916 

8 

9 

.44295 

.79527 

.45756 

.84352 

.47230 

.89503 

.48721 

.95011 

9 

10 

.44322 

.79604 

.45780 

.84435 

.47255 

.89591 

.48746 

.95106 

10 

11 

.44346 

.79682 

.45805 

.84518 

.47280 

.89680 

.48771 

.95201 

11 

12 

.44370 

.79761 

.45829 

.84601 

.47304 

.89769 

.48796 

.95296 

12 

13 

.44395 

.79839 

.45854 

.84685 

.47329 

.89858 

.48821 

.95392 

13 

14 

.44419 

.79917 

.45878 

.84768 

.47354 

.89948 

.48846 

.95487 

14 

15 

.44443 

.79995 

.45903 

.84852 

.47379 

.90037 

.48871 

.95583 

15 

16 

.44467 

.80074 

.45927 

.84935 

.47403 

.90126 

.48896 

.95678 

16 

17 

.44491 

.80152 

.45951 

.85019 

.47428 

.90216 

.48921 

.95774 

17 

18 

.44516 

.80231 

.45976 

.85103 

.47453 

.90305 

.48946 

.95870 

18 

19 

.44540 

.80309 

.46000 

.85187 

.47478 

.90395 

.48971 

.95966 

19 

20 

.44564 

.80388 

.46025 

.85271 

.47502 

.90485 

.48996 

.96062 

20 

21 

.44588 

.80467 

.46049 

.85355 

.47527 

.90575 

.49021 

.96158 

21 

22 

.44612 

.80546 

.46074 

.85439 

.47552 

.90665 

.49046 

.96255 

22 

23 

.44637 

.80625 

.46098 

.85523 

.47577 

.90755 

.49071 

.96351 

23 

24 

.44661 

.80704 

.46123 

.85608 

.47601 

.90845 

.49096 

.96448 

24 

25 

.44635 

.80783 

.46147 

.85692 

.47626 

.90935 

.49121 

.96544 

25 

26 

.44709 

.80862 

.46172 

.85777 

.47651 

.91026 

.49146 

.966-11 

26 

27 

.44734 

.80942 

.46196 

.85861 

.47676 

.91116 

.49171 

.967'38 

27 

28 

.44758 

.81021 

.46221 

.85946 

.47701 

.91207 

.49196 

.96835 

28 

29 

.44782 

.81101 

.46246 

.b6031 

.47725 

.91297 

.49221 

.96932 

29 

30 

.44806 

.81180 

.46270 

.86116 

.47750 

.91388 

.49246 

.97029 

30 

31 

.44831 

.81260 

.46295 

.86201 

.47775 

.91479 

.49271 

.97127 

31- 

32 

.44855 

.81340 

.46319 

.86286 

.47800 

.91570 

.49296 

.97224 

32 

33 

.44879 

.81419 

.46344 

.86371 

.47825 

.91661 

.49321 

.97322 

33 

34 

.44903 

.81499 

.46368 

.86457 

.47849 

.91752 

.49346 

.97420 

34 

35 

.44928 

.81579 

.46393 

.86542 

.47874 

.91844 

.49372 

.97517 

35 

36 

.44952 

.81659 

.46417 

.86627 

.47899 

.91935 

.49397 

.97615 

36 

37 

.44976 

.81740 

.46442 

.86713 

.47924 

.92027 

.49422 

.97713 

37 

38 

.45001 

.81820 

.46466 

.86799 

.47949 

.92118 

.49447 

.97811 

38 

39 

.45025 

.81900 

.46491 

.86885 

.47974 

.92210 

.49472 

.97910 

39 

40 

.45049 

.81981 

.46516 

.86990 

.47998 

.92302 

.49497 

.98008 

40 

41 

.45073 

.82061 

.46540 

.87056 

.48023 

.92394 

.49522 

.98107 

41 

42 

.45098 

.82142 

.46565 

.87142 

.48048 

.92486 

.49547 

.98205 

42 

43 

.45122 

.82222 

.46589 

.87229 

.48073 

.92578 

.49572 

.98304 

43 

44 

.45146 

.82303 

.46614 

.87315 

.48098 

.92670 

.49597 

.98403 

44 

45 

.45171 

.82384 

.46639 

.87401 

.48123 

.92762 

.49623 

.98502 

45 

46 

.45195 

.82465 

.46663 

.87488 

.48148 

.92855 

.49648 

.98601 

46 

47 

.45219 

.82546 

.46688 

.87574 

.48172 

.92947 

.49673 

.98700 

47 

48 

.45244 

.82627 

.46712 

.87661 

.48197 

.93040 

.49698 

.98799 

48 

49 

.45268 

.82709 

.46737 

.87748 

.48222 

.93133 

.49723 

.98899 

49 

50 

.45292 

.82790 

.46762 

.87834 

.48247 

.93226 

.49748 

.98998 

50 

51 

.45317 

.82871 

.46786 

.87921 

.48272 

.93319 

.49773 

.99098 

51 

52 

.4.5341 

.82953 

.46811 

.88008 

.48297 

.93412 

.49799 

.99198 

52 

53 

.45365 

.83034 

.46836 

.88095 

.4S322 

.93505 

.49824 

.99298 

53 

54 

.45390 

.83116 

.46860 

.88183 

.48347 

.93598 

.49849 

.99398 

54 

55 

.45414 

.83198 

.46885 

.88270 

.48372 

.93692 

.49874 

.99498 

55 

56 

.45439 

.83280 

.46909 

.88357 

.48396 

.93785 

.49899 

.99598 

56 

57 

-45463 

.83362 

.46934 

.88445 

.48421 

.93879 

.49924 

.99698 

57 

58 

.45487 

.83444 

.46959 

.88532 

.48446 

.93973 

.49950 

.99799 

58 

59 

.45512 

.83526 

.46983 

.88620 

.48471 

.94066 

.49975 

.99899 

59 

60 

.45536 

.83608 

.47008 

.88708  1 

.48496 

.94160 

.50000 

1.00000 

60 

261 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


60- 

61° 

62° 

63» 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.50000 

1.00000 

.51519 

1.06267 

.53053 

1.13005 

.54601 

1.20269 

0 

; 

.50025 

1.00101 

.51544 

1.06375 

.53079 

1.13122 

.54627 

1.20395 

1 

j 

.50050 

1.00202 

.51570 

1.06483 

.53104 

1.13239 

.54653 

1.20521 

2 

t 

.50076 

1.00303 

.51595 

1.06592 

.53130 

1.13356 

.54679 

1.20647 

i 

.50101 

1.00404 

.51621 

1.06701 

.53156 

1.13473 

.54705 

1.20773 

4 

5 

.50126 

1.00505 

.51646 

1.06809 

.53181 

1.13590 

.54731 

1.20900 

5 

6 

.50151 

1.00607 

.51672 

1.06918 

.53207 

1.13707 

.54757 

1.21026 

6 

r< 

.50176 

1.00708 

.51697 

1.07037 

.53233 

1.13825 

.54782 

1.21153 

8 

.50202 

1.00810 

.51723 

1.07137 

.53258 

1.13942 

.54808 

1.21280 

8 

1 

.50227 

1.0091:3 

.51748 

1.07246 

.53284 

1.14060 

.54834 

1.21407 

9 

1C 

.50252 

1.01014 

.51774 

1.07356 

.53310 

1.14178 

.54860 

1.21535 

10 

11 

.50277 

1.01116 

.51799 

1.07465 

.53336 

1.14296 

.54886 

1.21662 

11 

12 

.50303 

1.01218 

.51825 

1.07575 

.53361 

1.14414 

.54912 

1.21790  12 

13 

.50328 

1.01320 

.51850 

1.07685 

.53387 

1.14533 

.54938 

1.21918  113 

14 

.50353 

1.01422 

.51876 

1.07795 

.53413 

1.14651 

.54964 

1.22045 

14 

15 

.50378 

1.01525 

.51901 

1.07905 

.53439 

1.14770 

.54990 

1.22174 

15 

16 

.50404 

1.01628 

.51927 

1.08015 

.53464 

1.14889 

.55016 

1.22302 

10 

17 

.50429 

1.01730 

.51952 

1.08126 

.53490 

1.15008 

.55042 

1.22430 

17 

18 

.50454 

1.01833 

.51978 

1.08236 

.53516 

1.15127 

.55068 

1.22559 

18 

19 

.50479 

1.01936 

.52003 

1.08347 

.53542 

1.15246 

.55094 

1.22688 

19 

20 

.50505 

1.02039 

.52029 

1.08458 

.53567 

1.15366 

.55120 

1.22817 

20 

21 

.50530 

1.02143 

.52054 

1.08569 

.53593 

1.15485 

.55146 

1.22946 

21 

22 

.50555 

1.02246 

.52080 

1.08680 

.53619 

1.15605 

.66172 

1.23075 

00 

23 

.50581 

1.02349 

.52105 

1.08791 

.53645 

1.15725 

.55198 

1.23205 

23 

24 

.50606 

1.02453 

.52131 

1.08903 

.53670 

1.15845 

.55224 

1.23334 

24 

25 

.50631 

1.02557 

.52156 

1.09014 

.53696 

1.15965 

.55250 

1.23464 

25 

20 

.50656 

1.02661 

.52182 

1.09126 

.53722 

1.16085 

.55276 

1.23594 

20 

27 

.50682 

1.02765 

.52207 

1.09238 

.53748 

1.16206 

.55302 

1.23724 

27 

28 

.50707 

1.02869 

.52233 

1.09350 

.53774 

1.16326 

.55328 

1.23855 

88 

29 

.50732 

1.02973 

.52259 

1.09462 

.53799 

1.16447 

.55354 

1.23985 

•„".» 

30 

.50758 

1.03077 

.52284 

1.09574 

.53825 

1.16568 

.55380 

1.24116 

30 

31 

.50783 

1.03182 

.52310 

1.09686 

.53851 

1.16689 

.55406 

1.24247 

81 

32 

.50808 

1.03286 

.52335 

1.09799 

.53877 

1.16810 

.55432 

1.24378  132 

33 

.50834 

1.03391 

.52361 

1.09911 

.53903 

1.16932 

.55458 

1.24509 

3.3 

34 

.50859 

1.03496 

.52386 

1.10024 

.53928 

1.17053 

.55484 

1.24640 

84 

35 

.50884 

1.03601 

.52412 

1.10137 

.53954 

1.17175 

.55510 

1.24772 

35 

36 

.50910 

1.03706 

.52438 

1.10250 

.53980 

1.17297 

.55536 

1.24903 

83 

37 

.50935 

1.03811 

.52463 

1.10363 

.54006 

1.17419 

.55563 

1.25035 

87 

38 

.50960 

1.03916 

.52489 

1.10477 

.54032 

1.17541 

.55589 

1.25167 

3S 

39 

.50986 

1.04022 

.52514 

1.10590 

.54058 

1.17663 

.55615 

1.25300 

89 

40 

.51011 

1.04128 

.52540 

1.10704 

.54083 

1.17786 

.55641 

1.25432 

40 

41 

.51036 

1.04233 

.52566 

1.10817 

.54109 

1.17909 

.55667 

1.25565 

41 

42 

.51062 

1.04339 

.52591 

1.10931 

.54135 

1.18031 

.55693 

1.25697 

42 

43 

.51087 

1.04445 

.52617 

1.11045 

.54161 

1.18154 

.55719 

1.25830 

43 

44 

.51113 

1.04551 

.52642 

1.11159 

.54187 

1.18277 

.55745 

1.25063 

44 

45 

.51138 

1.04658 

.52668 

1.11274 

.54213 

1.18401 

.55771 

1.26097 

45 

46 

.51163 

1.04764 

.52694 

1.11388 

.54238 

1.18524 

.55797 

1.26230 

46 

47 

.51189 

1.04870 

.52719 

1.11503 

.54264 

1.18648 

.55823 

1.26364 

47 

48 

.51214 

1.04977 

.52745 

1.11617 

.5-1290 

1.18772 

.55849 

1.26498 

48 

49 

.51239 

1.05084 

.52771 

1.11732 

.54316 

1.18895 

.55876 

1.26632 

49 

50 

.51265 

1.05191 

.52796 

1.11847 

.54342 

1.19019 

.55902 

1.26766 

50 

51 

.51290 

1.05298 

.52822 

1.11963 

.54368 

1.19144 

.55928 

1.26900 

51 

52 

.51316 

1.05405 

.528-18 

1.12078 

.54394 

1.10268 

.55954 

1.27035 

52 

53 

.51341 

1.05512 

.52873 

1.12193 

.54120 

1.19393 

.55980 

1.27169 

63 

54 

.51366 

1.05619 

.52899 

1.12309  j 

.54446 

1.19517 

.56006 

1.27304 

54 

55 

.51392 

1.05727 

.52924 

1.12425 

.54471 

1.19642 

.56032 

1.27439 

55 

56 

.51417 

1.05835 

.52950 

1.12540 

.54497 

1.19767 

.56058 

1.27574 

66 

57 

.51443 

1.05942 

.52976 

1.12657 

.54523 

1.19892 

.56084 

1.27710 

57 

58 

.51468 

1.06050 

.53001 

1.12773 

.54549 

1.20018 

.56111 

1.27845 

53 

59 

.51494 

1.06158 

.53027 

1.12889 

.54575 

1.20143 

.56137 

1.27981 

59 

60 

.51519 

1.06267 

.53053 

1.13005 

.54601 

1.20269 

2,56163 

1.28117 

60 

262 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


6 

4° 

6 

5° 

6 

6' 

6 

7° 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.56163 

1.28117  ! 

.57738 

1.36620  ! 

.59326 

1.45859 

.60927 

1.55930 

1 

.56189 

.28253  I 

.57765 

1.3(5768 

.59353 

1.46020 

.60954 

1.56106 

2 

.56215 

.28390  ! 

.57791 

1.36916  ! 

.59379 

1.46181 

.60980 

.56282 

3 

.56241 

.28526  i 

.57817 

1.37064  i 

.59406 

1.46342  ; 

.61007 

.56458 

4 

.56267 

.28663  i 

.57844 

1.37212 

.59433 

1.46504  1 

.61034 

.56634 

5 

.56294 

.28800  I 

.57870 

1.37361  ! 

.59459 

1.46665  i 

.61061 

.56811 

6 

.56320 

.28937 

.57896 

1.37509 

.59486 

1.46827 

.61088 

.56988 

7 

.56346 

.29074 

.57923 

1.37658 

.59512 

1.46989 

.61114 

.57165 

8 

.56372 

.29211 

.57949 

1.37808 

.59539 

1.47152 

.61141 

.57342 

9 

.56398 

.29349  i 

.57976 

1.37957 

.59566 

1.47314 

.61168 

.57520 

10 

.56425 

.29487 

.58002 

1.3810? 

.59592 

1.47477 

.61195 

.57698 

11 

.56451 

.20625 

.58028 

1.38256 

.59619 

1.47640 

.61222 

.57876 

12 

.50477 

•29763 

.58055 

1.38406 

.59645 

1.47804 

.61248 

.58054 

13 

.56503 

.29901 

.58081 

1.38556 

.59672 

1.47967 

.61275 

.58233 

14 

.56529 

.30040 

.58108 

1.38707 

.59699 

1.48131 

.61302 

.58412 

15 

.56555 

.30179 

.58134 

1.38857 

.59725 

1.48295 

.61329 

.58591 

16 

.56582 

.30318  ! 

.58160 

1.39008 

.59752 

1.48459  ! 

.61356 

.58771 

17 

.56608 

.30457  i 

.58187 

1.39159 

.59779 

1.48624 

.61383 

.58950 

18 

.56634 

.30596  i 

.58213 

1.39311 

.59805 

1.48789 

.61409 

.59130 

19 

.500(50 

.30735  ! 

.58240 

1.39462 

.59832 

1.48984 

.61436 

.59311 

20 

.56687 

.30875 

.58266 

1.39614 

.59859 

1.49119 

.61463 

59491 

21 

.56713 

.31015 

.58293 

1.39766 

.59885 

1.49284 

.61490 

.59672 

22 

.50739 

.31155 

.58319 

1.39918 

.59912 

1.49450 

.61517 

.59853 

23 

.56765 

.31295 

.5*345 

1.40070 

.59938 

1.49616 

.61544 

.60035 

21 

.56791 

.31436 

.58372 

1.40222 

.59965 

1.49782 

.61570 

.60217 

25 

.56818 

.31576  I 

.58398 

1.40375 

.59992 

1.49948 

.61597 

.60399 

20 

.56844 

.31717 

.58425 

1.40528 

.60018 

1.50115 

.61624 

.60581 

27 

.56870 

.31858 

.58451 

1.40681 

.60045 

1.50282 

.61651 

.60763 

& 

.56896 

.31999 

.58478 

1.40835 

.60072 

1.50449 

.61678 

.60946 

29 

.56923 

.32140 

.58504 

1.40988 

.60098 

1.50617 

.61705 

.61129 

30 

.56949 

.32282 

.58531 

1.41142 

.60125 

1.50784 

.61732 

.61313 

31 

.56975 

.32424 

.58557 

1.41296 

.60152 

1.50952 

.61759 

.61496 

32 

.57001 

.32566 

.58584 

1.41450 

.60178 

1.51120 

.61785 

.61680 

88 

.57028 

.32708 

.58610 

1.41605 

.60205 

1.51289 

.61812 

.61864 

31 

.57054 

.32850 

.58637 

1.41760 

.60232 

1.51457 

.61839 

.62049 

36 

.57080 

.32993 

.58663 

1.41914 

.60259 

1.51626 

.61866 

.62234 

86 

.57106 

.33135 

.58690 

1.42070 

.60285 

1.51795 

.61893 

.62419 

37 

.57133 

.33278 

.58716 

1.42225 

.60312 

1.51965 

.61920 

.62604 

:js 

.57159 

.33422 

.58743 

1.42380 

.60339 

1.52134  i 

.61947 

.62790 

99 

.57185 

1.33565 

.58769 

1.42536 

.60365 

1.52304 

.61974 

.62976 

40 

.57212 

1.33708 

.58796 

1.42692 

.60392 

1.52474  i 

.62001 

.63162 

41 

.57238 

1.33852 

.58822 

1.42848 

.60419 

1.52645 

.62027 

.63348 

42 

.57204 

1.33996 

.58849 

1.43005 

.60445 

1.52815 

.62054 

.63535 

43 

.57291 

1.34140 

.58875 

1.43162 

.604?2 

1.52986 

.62081 

.63722 

44 

.57317 

1.34284 

.58902 

1.43318 

.60499 

1.53157 

.62108 

.63909 

45 

.57343 

1.34429 

.58928 

1.43476 

.60526 

1.53329  ! 

.62135 

.64G97 

46 

.57369 

1.34573 

.58955 

1.43633 

.60552 

1.53500 

.62162 

.64285 

47 

.57396 

1.34718 

.58981 

1.43790 

.60579 

1.53672 

.62189 

.64473 

48 

.57422 

1.34863 

.59008 

1.43948 

.60606 

1.53845 

.62216 

.64662 

49 

.57448 

1.35009 

.59034 

1.44106 

.60633 

1.54017 

.62243 

.64851 

50 

.57475 

1.35154 

.59061 

1.44264 

.60659 

1.54190 

.62270 

.65040 

51 

.57501 

1.35300 

.59087 

1.44423 

.60686 

1.54363 

.62297 

.65229 

52 

.57527 

1.35446 

.59114 

1.44582 

.60713 

1.54536 

.62324 

.65419 

58 

.57554 

1.35592 

.59140 

1.44741 

.60740 

1.54709 

.62351 

.65609 

54 

.57580 

1.35738 

.59167 

1.44900 

.60766 

1.54883 

.62378 

.65799 

55 

.57606 

1.35885 

.59194 

1.45059 

.60793 

1.5505? 

.62405 

.65989 

50 

.57633 

1.36031 

.59220 

1.45219 

.60820 

1.55231 

.62431 

.66180 

57 

.57659 

1.36178 

.59247 

1.45378 

.60847 

1.55405 

.62458 

.66371 

58 

.57685 

1.36325 

.59273 

1.45539 

.60873 

1.55580 

.62485 

.66563 

59 

.57712 

1.36473 

.59300 

1.45699 

.60900 

1.55755 

.62512 

.66755 

GO 

.57738 

1.36620 

.59326 

1.45859 

.60927 

1.55930 

.62539 

.66947 

263 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


68- 

69° 

70° 

71° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0  .62539 

1.66947 

.64163 

1.79043 

.65798 

1.92380 

.67443 

2.07155 

Q 

1  .62566 

1.67139 

.64190 

1.79254 

.65825 

1.92614 

.67471 

2.07415 

• 

2  .62593 

1.67332 

.64218 

1.79466 

.65853 

1.92849 

.67498 

2.07675 

3  .62620 

1.67525 

.64245 

1.79679 

.65880 

1.93083 

.67526 

2.07936 

4  .62647 

1.67718 

.64272 

1.79891 

.65907 

1.93318 

.67553 

2.08197 

^ 

5  .62674 

1.67911 

.64299 

1.80104 

.65935 

1.93554 

.677)81 

2.08459 

( 

6  .62701 

1.68105 

.64326 

1.80318 

.65962 

1.93790 

.67608 

2.08721 

( 

7  .62728 

1.68299 

.64353 

1.80531 

.65989 

1.94026 

.67636 

2.08983 

t 

8  .62755 

1.68494 

.64381 

1.80746 

.66017 

1.94263 

.67663 

2.09246 

| 

9  .62782 

1.68689 

.64408 

1.80960 

.66044 

1.94500 

.67691 

2.09510 

| 

10  .62809 

1.68884 

.64435 

1.81175 

.66071 

1.94737 

.67718 

2.09774 

10 

11  .62836 

1.69079 

.64462 

1.81390 

.66099 

1.94975 

.67746 

2.10038 

11 

12  .62863 

1.69275 

.64489 

1.81605 

.66126 

1.95213 

.67773 

2.10303 

12 

3  .62890 

1.69471 

.64517 

1.81821 

.66154 

1.95452 

.67801 

2.10568 

13 

4  .62917 

1.69667 

.64544 

1.82037 

.66181 

1.95691 

.67829 

2.10834 

14 

5  .62944 

1.69864 

.64571 

1.82254 

.66208 

1.95931 

.67856 

2.11101 

15 

6  .62971 

1.70061 

.64598 

1.82471 

.66236 

1.96171 

.67884 

2.11367 

16 

7  .62998 

1.70258 

.64625 

1.82688 

.66263 

1.96411 

.67911 

2.11635 

17 

8  .63025 

1.70455 

.64653 

1.82906 

.66290 

1.96652 

.67939 

2.11903 

18 

9  .63052 

1.70653 

.64680 

1.83124 

.66318 

1.96893 

.C7966 

2.12171 

19 

20  .63079 

1.70851 

.64707 

1.83342 

.66345 

1.97135 

.67994 

2.12440 

20 

21  .63106 

1.71050 

.64734 

1.83561 

.66373 

1.97377 

.68021 

2.12709 

21 

22  .63133 

1.71249 

.64761 

1.83780 

.66400 

1.97619 

.68049 

2.12979 

00 

23  .63161 

1.71448 

.64789 

1.83999 

.66427 

1.97862 

.68077 

2.13249 

23 

24  .63188 

1.71647 

.64816 

1.84219 

.66455 

1.98106 

.68104 

2.13520 

24 

25  .63215 

1.71847 

.64843 

1.84439 

.66482 

1.98349 

.68132 

2.13791 

25 

26  .63242 

1.72047 

.64870 

1.84659 

.66510 

1.9S594 

.68159 

2.14063 

26 

27  .63269 

1.72247 

.64898 

1.84880 

.66537 

1.98838 

.68187 

2.14335 

27 

28  .63296 

1.72448 

.64925 

1.85102 

.66564 

1.99083 

.68214 

2.14608 

23 

29  .63323 

1.72649 

.64952 

1.85323 

.66592 

1.99329 

.68242 

2.14881 

Of 

30  .63350 

1.72850 

.64979 

1.85545 

.66619 

1.99574 

.68270 

2.15155 

30 

31  .63377 

1.73052 

.65007 

1.85767 

.66647 

1.99821 

.68297 

2.15429 

31 

32  .63404 

1.73254 

.65034 

1.85990 

.66674 

2.00067 

.68325 

2.15704 

32 

33  .63431 

1.73456 

.65061 

1.86213 

.66702 

2.00315 

.68352 

2.15979 

33 

34  .63458 

1.73659 

.65088 

1.86437 

.66729 

2.00562 

.68380 

2.16255 

34 

35  .63485 

1.73862 

.65116 

1.86661 

.66756 

2.00810 

.68408 

2.16531 

35 

36  .63512 

1.74065 

.65143 

1.86885 

.66784 

2.01059 

.68435 

2.16808 

80 

37  .63539 

1.74269 

.65170 

1.87109 

.66811 

2.01-308 

.68463 

2.17085 

37 

38  .63566 

1.74473 

.65197 

1.87334 

.66839 

2.01557 

.68490 

2.17363 

38 

39  .63594 

1.74677 

.65225 

1.87560 

.66866 

2.01807 

.68518 

2.17641 

39 

40  .63621 

1.74881 

.65252 

1.87785 

.66894 

2.02057 

.68546 

2.17920 

40 

41  .63648 

1.75086 

.65279 

1.88011 

.66921 

2.02308 

.68573 

2.18199 

41 

42  .63875 

1.75292 

.65306 

1.88238 

.66949 

2.02559 

.68601 

2.18479 

42 

43  .63702 

1.75497 

.65334 

1.88465 

.66976 

2.02810 

.68628 

2.18759 

43 

44  .63729 

1.75703 

.65361 

1.88692 

.67003 

2.03062 

.68656 

2.19040 

44 

45  .63756 

1.75909 

.65388 

1.88920 

.67031 

2.03315 

.68684 

2.19322 

15 

46  .63783 

1.76116 

.65416 

1.89148 

.67058 

2.03568 

.68711 

2.19604 

to 

47  .63810 

1.76323 

.65443 

1.89376 

.67'086 

2.03821 

.68739 

2.19886 

[7 

48  .63838 

1.76530 

.65470 

1.89605 

.67113 

2.04075 

.68767 

2.20169 

48 

49  .63865 

1.76737 

.65497 

1.89834 

.67141 

2.04329 

.68794 

2.20453 

49 

30  .63892 

1.76945 

.65525 

1.90063 

.67168 

2.04584 

.68822 

2.20737 

50 

51  .63919 

1.77154 

.65552 

1.90293 

.67196 

2.04839 

.68849 

2.21021 

51 

32  .63946 

1.77362 

.65579 

1.90524 

.67223 

2.05094 

.68877 

2.21306 

53  .63973 

1.77571  i 

.65607 

1.90754 

.67251 

2.05350 

.68905 

2.21592  53 

54  .64000 

1  77780 

.65634 

1.90986 

.67278 

2.05607 

.68932 

2.21878  |54 

55  .64027 

1.77990 

.65661 

1.91217 

.67306 

2.05864 

.68960 

2.22165 

55 

56  .64055 

1.78200 

.65689 

1.91449 

.67333 

2.06121 

.68988 

2.22452 

56 

57  .64082 

1.78410  ( 

.65716 

1.91681 

.67361 

2.06379 

.69015 

2.22740 

57 

58  .64109 

1.78621 

.65743 

1.91914 

.67388 

2.06637 

.69043 

2.23028 

58 

59  .641--J6 

1.78832 

.65771 

1.92147 

.67416 

2.06896 

.69071 

2.23317 

59 

30   64163 

1.79043  I 

.65798 

1.92380 

.67443 

2.07155 

.69098 

2.23607 

60 

264 


TABLE  Xm.— VERSINES  AND  EXSECANTS. 


' 

72"' 

73° 

74° 

75° 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec.  • 

Vers. 

Exsec. 

0 

.69098 

2.23607 

.70763 

2.42030 

.72436 

2.62796 

.74118 

2.86370 

1 

.69126 

2.23897 

.70791 

2.42356 

.72464 

2.63164 

.74146 

2.86790 

2 

.69154 

2.24187 

.70818 

2.42683 

.72492 

2.63533 

.74174 

2.87211 

3 

.69181 

2.24478 

.70846 

2.43010 

.72520 

2.63903 

.74202 

2.87633 

4 

.69209 

2.24770 

.70874 

2.43337 

.72548 

2.64274 

.74231 

2.88056 

5 

.69237 

2.25062 

.70902 

2.43666 

.72576 

2.64645 

.74259 

2.88479 

6 

.69264 

2.25355 

.70930 

2.43995 

.72604 

2.65018 

.74287 

2.88904 

7 

.69292 

2.25648 

.70958 

2.44324 

.72632 

2.65391 

.74315 

2.89330 

8 

.69320 

2.25942 

.70985 

2.44655 

.72660 

2.65765 

.74343 

2.89756 

9 

.69347 

226237 

.71013 

2.44986 

.72688 

2.66140 

.74371 

2.90184 

10 

.69375 

2.26531 

.71041 

2.45317 

.72716 

2.66515 

.74399 

2.90G13 

11 

.69403 

2.26S27 

.71069 

2.45650 

.72744 

2.66892 

.74427 

2.91042 

13 

.69430 

2.27123 

.71097 

2.45983 

.72772 

2.67269 

.74455 

2.91473 

13 

.69458 

2.27420 

.71125 

2.46316 

.72800 

2.67647 

.74484 

2.91904 

14 

.69486 

2.27717 

.71153 

2.46651 

.72828 

2.68025 

.74512 

2.92337 

15 

.69514 

2.28015 

.71180 

2.46986 

.72856 

2.68405 

.74540 

2.92770 

1(3 

.69541 

2.28313 

.71208 

2.47321 

.72884 

2.68785 

.74568 

2.93204 

17 

.69569 

2.28612 

.71236 

2.47658 

.72912 

2.69167 

.74596 

2.93640 

18 

.69597 

2.28912 

.71264 

2.47995 

.72940 

2.69549 

.74624 

2.94076 

19 

.69624 

2.29212 

.71292 

2.48333 

.72968 

2.69931 

.74652 

2.94514 

20 

.69652 

2.29512 

.71320 

2.48671 

.72996 

2.70315 

.74680 

2.9495JJ 

21 

.69680 

2.29814 

.71348 

2.49010 

.73024 

2.70700 

.74709 

2.95392 

22 

.69708 

2.30115 

.71375 

2.49350 

.73052 

2.71085 

.74737 

2.95832 

23 

.69735 

2.30418 

.71403 

2.49691 

.73080 

2.71471 

.74765 

2.96274 

24 

.69763 

2.30721 

.71431 

2.50032 

.73108 

2.71858 

.74793 

2.96716 

25 

.69791 

2.31024 

.71459 

2.50374 

.73136 

2.72246 

.74821 

2.97160 

26 

.69818 

2.31328 

.71487 

2.50716 

.73164 

2.72635 

.74849 

2.97604 

27 

.69846 

2.31633 

.71515 

2.51060 

.73192 

2.73024 

.74878 

2.98050 

28 

.69874 

2.31939 

.71543 

2.51404 

.73220 

2.73414 

.74906 

2.98497 

29 

.69902 

2.32244 

.71571 

2.51748 

.73248 

2.73806 

.74934 

2.98944 

30 

.69929 

2.32551 

.71598 

2.52094 

.73276 

2.74198 

.74962 

2.99393 

31 

.69957 

2.32858 

.71626 

2.52440 

.73304 

2.74591 

.74990 

2.99843 

32 

.69985 

2.33166 

.71654 

2.52787 

•  (•>>•!'* 

2.74984 

.75018 

3.00293 

33 

.70013 

2.33474 

.71682 

2.53134 

.73360 

2.75379 

.75047 

3.00745 

34 

.70040 

2.33783 

.71710 

2.53482 

.73388 

2.75775 

.75075 

3.01198 

35 

.70068 

2.34092 

.71738 

2.53831 

.73416 

2.76171 

.75103 

3.01652 

36 

.70096 

2.34403 

.71766 

2.54181 

.73444 

2.76568 

.75131 

3.02107 

37 

.70124 

2.34713 

.71794 

2.54531 

.73472 

2.76966 

.75159 

3.02563 

38 

.70151 

2.35025 

.71822 

2.54883 

.73500 

2.77365 

.75187 

3.03020 

39 

.70179 

2.35336 

.71850 

2.55235 

.73529 

2.77765 

.75216 

3.03479 

40 

.70207 

2.35649 

.71877 

2.55587 

.73557 

2.78166 

.75244 

3.03938 

41 

.70235 

2.35962 

.71905 

2.55940 

.73585 

2.78568 

.75272 

3.04398 

42 

.70263 

2.36276 

.71933 

2.56294 

.73613 

2.78970 

.75300 

3.04860 

43 

.70290 

2.36590 

.71961 

2.56649 

.73641 

2.79374 

.75328 

3.05322 

44 

.70318 

2.36905 

.71989 

2.57005 

.73669 

2.79778 

.75356 

3.05786 

45 

.70346 

2.37221 

.72017 

2.57361 

.73697 

2.80183 

.75385 

3.06251 

46 

.70374 

2.37537 

.72045 

2.57718 

.73725 

2.80589 

.75413 

3.06717 

47 

.70401 

2.37854 

.72073 

2.58076 

.73753 

2.80996 

.75441 

3.07184 

48 

.70429 

2.38171 

.72101 

2.58434 

.73781 

2.81404 

.75469 

3.07652 

49 

.70457 

2.38489 

.72129 

2.58794 

.73809 

2.81813 

.75497 

3.08121 

50 

.70485 

2.38808 

.72157 

2.59154 

.73837 

2.82223 

.75526 

3.08591 

51 

.70513 

2.39128 

.72185 

2.59514 

.73865 

2.82633 

.75554 

3.09063 

52 

.705-10 

2.39448 

.72213 

2.59876 

.73893 

2.83045 

.75582 

3.09535 

53 

.70568 

2.39768 

.72241 

2.60238 

.73921 

2.83457 

.75610 

3.10009 

54 

.70596 

2.40089 

.72269 

2.60601 

.73950 

2.83871 

.75689 

3.10484 

55 

.70624 

2.40411 

.72296 

2.60965 

.73978 

2.84285 

.75667 

3.10960 

56 

.70652 

2.40734 

.72324 

2.61330 

.74006 

2.84700 

.75695 

3.11437 

57 

.70679 

2.41057 

.72352 

2.61695 

.74034 

2.85116 

.75723 

3.11915 

58 

.70707 

2.41381 

.72380 

2.62061 

.74062 

2.85533 

.75751 

3.12394 

59 

.70735 

2.41705 

.72408 

2.62428 

.74090 

2.85951 

.75780 

3.12875 

60 

.70763 

2.42030 

.72436 

2.62796 

.74118 

2.86370 

.75808 

3.13357 

265 


.TABLE  XIII.— VERSINES  AND  EXSECANTS. 


7 

6° 

7 

7° 

7 

8° 

7 

9° 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

j  Vers. 

Exsec. 

0 

.75808 

3.13357 

.77505 

3.44541 

.79209 

3.80973 

.80919 

4.24084 

0 

1 

.75836 

3.13839 

.77'533 

3.45102 

.79237 

3.81633 

.80948 

4.24870 

1 

2 

75864 

3.14323 

.77562 

3.45664 

.79266 

3.82294 

.80976 

4.25658 

2 

3 

7'5892 

3.14809 

.77590 

3.46228 

.79294 

3.82956 

.81005 

4.26448 

3 

4 

75921 

3.15295 

.77618 

3.46793 

.79323 

3.83621 

.81033 

4.27241 

4 

5 

75949 

3.15782 

.77647 

3.47360 

.79351 

3.84288 

.81062 

4.28030 

5 

G 

75977 

3.16271 

.77675 

3.47928 

.79380 

3.84956 

.81090 

4.28833 

6 

7 

76005 

3.16761 

.77703 

8.48498 

.79408 

3.85627 

.81119 

4.29634 

7 

8 

76034 

3.17252 

.77732 

3.49069 

.79437 

3.86299 

.81148 

4.30436 

8 

9 

76062 

3.17744 

.77760 

3.49642 

.79465 

3.86973 

.81176 

4.31241 

g 

10 

76090 

3.18238 

.77788 

3.50216 

.79493 

3.87649 

.81205 

4.32049 

10 

11 

.76118 

3.18733 

.77817 

3.50791 

.79522 

3.88327 

.81233 

4.32P59 

11 

12 

.76147 

3.19228 

.77845 

3.51368 

.79550 

3.89007 

.81262 

4.33(171 

12 

13 

.76175 

3.19725 

.77874 

3.51947 

.79579 

3.89689 

.81290 

4.34480 

18 

14 

.76203 

3.20224 

.77902 

3.52527 

.79607 

3.90373 

.81319 

4.35304 

14 

15 

.76231 

3.20723 

.77930 

3.53109 

.79636 

3.91058 

.81348 

4.36124 

Ifi 

16 

.76260 

3.21224 

.77959 

3.53692 

.79664 

3.91746 

.81376 

4.36947 

18 

17 

.76288 

3.21726 

.77987 

3.54277 

.79693 

3.92436 

1  .81405 

4.37772 

17 

18 

.76316 

3.22229 

.78015 

3.54863 

.79721 

3.93128 

.81433 

4.38600 

18 

19 

.76344 

3.22734 

.78044 

3.55451 

.79750 

3.93821 

.81403 

4.3943') 

19 

20 

.76373 

3.23239 

.78072 

3.56041 

.79778 

3.94517 

.81491 

4.40263 

20 

21 

.76401 

3.23746 

.78101 

3.56632 

.79807 

3.95215 

.81519 

4.41099 

21 

22 

.76429 

3.24255 

.78129 

3.57224 

.79835 

3.95914 

.81548 

4.41937 

O-i 

23 

.76453 

3.24764 

.78157 

3.57819 

.79864 

3.S6616 

.81576 

4.42778 

28 

24 

.76486 

3.25275 

.78186 

3.58414 

.79892 

3.97320 

.81605 

4.43022 

x!4 

25 

.76514 

3.25787 

.78214 

3.59012 

.79921 

3.98025 

.81633 

4.41408 

,'.'.-> 

2(5 

.76542 

3.26300 

.78242 

3.59611 

.7994!) 

3.98733 

.81662 

4.45317 

26 

27 

.70571 

3.26814 

.78271 

3.G0211 

.79973 

3.99443 

1  .81691 

4.  46?  09 

87 

28 

.76599 

3.27330 

.78299 

3.60813 

.80006 

4.00155 

i  .81719 

4.47023 

L'8 

29 

.76627 

3.27847 

.78328 

3.61417 

.80035 

4.00869 

1  .81748 

4.47'881 

28 

30 

.76655 

3.28366 

.78356 

3.62023 

.80063 

4.01585 

.81776 

4.48740 

*u 

31 

.76684 

3.28885 

.7838-1 

3.62630 

.80092 

4.02303 

i  .81805 

4.49603 

81 

32 

.76712 

3.29406 

.78413 

3.63238 

.80120 

4.03024 

!  .81834 

4.50408 

3.^ 
A 

33 

.76740 

3.29929 

.78441 

3.63849 

.80149 

4.03746 

.81862 

4.51387 

88 

34 

.76769 

3.30452 

.78470 

3.61461 

'.80177 

4.04471 

.81891 

4.52208 

34 

35 

.76797 

3.30977 

.78498 

3.65074 

.80206 

4.05197 

.81919 

4.53081 

35 

36 

.76825 

3.31503 

.78526 

3.65690 

.80234 

4.05926 

.81948 

4.53958 

88 

37 

.76854 

3.32031 

.78555 

3.66307 

.80263 

4.06657 

.81977 

4.54837 

3V 

38 

.76882 

3.32560 

.78583 

3.60925 

.80291 

4.07390 

.82005 

4.55720 

88 

39 

.76910 

3.33090 

.78612 

3.67545 

.80320 

4.08125 

.82034 

4.56605 

39 

40 

.76938 

3.83622 

.78640 

3.68167 

.80348 

4.08863 

.82063 

4.57493 

40 

41 

.76967 

3.34154 

.78669 

3.68791 

.80377 

4.09602 

.82091 

4.58383 

41 

42 

.76995 

3.34689 

.78697 

3.69417 

.80405 

4.10344 

.82120 

4.59277 

42 

43 

.77023 

3.35224 

.78725 

3.7'0044 

.80434 

4.11088 

.82148 

4.6017'4 

43 

44 

.77052 

3.35761 

.78754 

3.70673 

.80462 

4.11835 

.82177 

4.61073 

44 

45 

.77080 

3.36299 

.78782 

3.71303 

.80491 

4.12583 

.82206 

4.61976 

45 

46 

.77108 

3.36839 

.78811 

3.71935 

.80520 

4.13334 

.82234 

4.62881 

40 

47 

.77137 

3.37380 

.78839 

3.72569 

.80548 

4.14087 

.82263 

4.63790 

47 

48 

.77165 

3.37923 

.78868 

3.73205 

.80577 

4.14842 

.82292 

4.64701 

48 

49 

.77193 

3.38466 

.78896 

8.  73843 

.80605 

4.15599 

.82320 

4.65010 

49 

60 

.77222 

3.39012 

.78924 

3.74482 

.80634 

4.16359 

.82349 

4.G0533 

50 

51 

.77250 

3.39558 

.78953 

3.75123 

.80669 

4.17121 

.82377 

4.67454 

51 

52 

.77278 

3.40106 

.78981 

3.75rG6 

.80691 

4.17886 

.82406 

4.08377 

52 

53 

.77307 

3.40656 

.79010 

3.76411 

.80719 

4.18652 

.82435 

4.69304 

53 

54 

.77335 

3.41206 

.79038 

3.77057 

.80748 

4.19421 

.82463 

4.70234 

54 

55 

.77363 

3.41759 

.79067 

3  .  77705 

.80776 

4.20193 

.82492 

4.71166 

55 

56 

.77392 

3.42312 

.79095 

3.78355 

.80805 

4.20966 

.82521 

4.72102 

56 

57 

.77420 

3.42867 

.79123 

3.79007 

.80833 

4.21742 

82549 

4.73041 

57 

58 

.77448 

3.43424 

.79152 

3.79661 

.80862 

4.22521 

.82578 

4.73983 

58 

59 

.77477 

3.43982 

.79180 

3.80316 

.80891 

4.23301 

.82607 

4.74929 

59 

60 

.77505 

3.44541 

.79209 

3.80973 

.80919 

4.24084 

.82035 

4.75877 

60 

266 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


/ 

80° 

81°          82°          83° 

! 

/ 

Vers. 

Exsec. 

Vers. 

Exsec.   Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.82635 

4.75877 

.84357 

5.39245 

.86083 

6.18530 

.87813 

7.205*1 

0 

1 

.82664 

4.76829 

.84385 

5.40422 

.86112 

6.20020 

.87842 

7.22500 

2 

.82692 

4.77784 

.84414 

5.41602 

.86140 

6.21517 

.87871 

7.24457 

& 

3 

.82721 

4.78742 

.84443 

5.42787 

.86169 

6.23019 

.87900 

7.26425 

3 

4 

.82750 

4.79703 

.84471 

5.43977 

.86198 

6.24529 

.87929 

7.28402 

4 

5 

.82778 

4.80667 

.84500 

5.45171 

.86227 

6.26044 

.87957 

7.36388 

5 

6 

.82807 

4.81635 

.84529 

5.46369 

.86256 

6.27566 

.87986 

7.32384 

6 

7 

.82836 

4.82606 

.84558 

5.47572 

.86284 

6.29095 

.88015 

7.34390 

7 

8 

.82864 

4.83581 

.84586 

5.48779 

.86313 

6.30630 

.88044 

7.36405 

8 

9 

.82893 

4.84558 

.84615 

5.49991 

.86342 

6.32171 

.88073 

7.38431 

9 

10 

.82922 

4.85539 

.84644 

5.51208 

.86371 

6.33719 

.88102 

7.40466 

10 

11 

.82950 

4.86524 

.84673 

5.52429 

.86400 

6.35274 

.88131 

7.42511 

11 

12 

.82979 

4.87511 

.84701 

5.53655 

.86428 

6.36835 

.88160 

7.44566 

12 

13 

.83003 

4.88502 

.84730 

5.54886 

.86457 

6.38403 

.88188 

7.46632 

13 

14 

.83036 

4.89497 

.84759 

5.56121 

.86486 

6.39978 

.88217 

7.48707 

14 

15 

.83065 

4.90495 

.84788 

5.57361 

.86515 

6.41560 

.88246 

7.50793 

15 

16 

.83094 

4.91496 

.84816 

5.58606 

.86544 

6.43148 

.88275 

7.52889 

16 

17 

.83122 

4.92501 

.84845 

5.59855 

.86573 

6.44743 

.88304 

7.54996 

17 

18 

.83151 

4.93509 

.84874 

5.61110 

.86601 

6.46346 

.88333 

7.57113 

18 

19 

.83180 

4.94521 

.84903 

5.62369 

.86630 

6.47955 

.88362 

7.59241 

19 

20 

.83208 

4.95536 

.84931 

5.63633 

.86659 

6.49571 

.88391 

7.61379 

20 

21 

.83237 

4.96555 

.84960 

5.64902 

.86688 

6.51194 

.88420 

7.63528 

21 

22 

.83266 

4.97577 

.84989 

5.66176 

.86717 

6.52825 

.88448 

7.65688 

22 

23 

.83294 

4.98603 

.85018 

5.67454 

.86746 

6.54462 

.88477 

7.67859 

23 

24 

.83323 

4.99633 

.85046 

5.68738 

.86774 

6.56107 

.88506 

7.70041 

24 

25 

.83352 

5.00666 

.85075 

5.70027 

.86803 

6.57759 

.88535 

7.72234 

25 

26 

.83380 

5.01703 

.85104 

5.71321 

•  .86832 

6.59418 

.88504 

7.74438 

26 

27 

.83409 

5.02743 

.85133 

5.72620 

.86861 

6.61085 

.88593 

7.76653 

27 

28 

.83438 

5.03787 

.85162 

5.73924 

.86890 

6.62759 

.88622 

7.78880 

28 

29 

.83467 

5.04834 

.85190 

5.75233 

.86919 

6.64441 

.88651 

7.81118 

29 

30 

.83495 

5.05886 

.85219 

5.76547 

.86947 

6.66130 

.88680 

7.83367 

30 

31 

.83524 

5.06941 

.85248 

5.77866 

.86976 

6.67826 

.88709 

7.85628 

31 

32 

.83553 

5.08000 

.85277 

5.79191 

.87005 

6.69530 

.88737 

7.87901 

32 

33 

.83581 

5.09062 

.85305 

5.80521 

.87034 

6.71242 

.88766 

7.90186 

33 

34 

.83610 

5.10129 

.85334 

5.81856 

.87063 

6.72962 

.88795 

7.92482 

34 

35 

.83639 

5.11199 

.85363 

5.83196 

.87092 

6.74689 

.88824 

7.94791 

35 

36 

.83667 

5.12273 

.85392 

5.84542 

.87120 

6.76424 

.88853 

7.97111 

36 

37 

.83696 

5.13350 

.85420 

5.85893 

.87149 

6.78167 

.88882 

7.99444 

37 

38 

.83725 

5.14432 

.85449 

5.87250 

.87178 

6.79918 

.88911 

8.01788 

38 

39 

.83754 

5.15517 

.85478 

5.88612 

.87207 

6.81677 

.88940 

8.04146 

39 

40 

.83782 

5.16607 

.85507 

5.89979 

.87236 

6.83443 

.88969 

8.06515 

40 

41 

.83811 

5.17700 

.85536 

5.91352 

.87265 

6.85218 

.88998 

8.08897 

41 

42 

.83840 

5.18797 

.85564 

5.92731 

.87294 

6.87001 

.89027 

8.11292 

42 

43 

.83868 

5.19896 

.85593 

5.94115 

.87322 

6.88792 

.89055 

8.13699 

43 

44 

.83897 

5.21004 

.85622 

5.95505 

.87351 

6.90592 

.89084 

8.16120 

44 

45 

.83926 

5.22113 

.85651 

5.96900 

.87380 

6.92400 

.89113 

8.18553 

45 

46 

.83954 

5.23226 

.85680 

5.98301 

.87409 

6.94216 

.89142 

8.20999 

46 

47 

.83983 

5.24343 

.85708 

5.99708 

.87438 

6.96040 

.89171 

8.23459 

47 

48 

.84012 

5.25464 

.85737 

6.01120 

.87467 

6.97873 

.89200 

8.25931  148 

49 

.84041 

5.26590 

.85766 

6.02538 

.87496 

6.99714 

.89239 

8.28417  49 

50 

.84069 

5.27719 

.85795 

6.03962 

.87524 

7.01565 

.89258 

8.30917 

50 

51 

.84098 

5.28853 

.85823 

6.05392 

.87553 

7.03423 

.89287 

8.33430 

51 

52 

.84127 

5.29991 

.85852 

6.06828 

.87582 

7.05291 

.89316 

8.35957 

52 

53 

.84155 

5.31133 

.85881 

6.08269 

.87611 

7.07167 

.89345 

8.38497 

53 

54 

.84184 

5.32279 

.85910 

6.09717 

.87640 

7.09052 

.89374 

8.41052 

54 

55 

.84213 

5.33429 

.85939 

6.11171 

.87669 

7.10946 

.89403 

8.43620 

55 

56 

.84242 

5.34584 

.85967 

6.12630 

.87698 

7.12849 

.89431 

8.46203 

56 

57 

.84270 

5.35743 

.85996 

6.14096 

.87726 

7.14760 

.89460 

8.48800 

57 

58 

.84299 

5.36906 

.86025 

6.15568 

.87755 

7.16681 

.89489 

8.51411 

58 

50 

.84328 

5.38073 

.86054 

6.17046 

.87784 

7.18612 

.89518 

8.54037 

59 

60 

.84357 

5.39245 

.86083 

6.18530 

.87813 

7,20551 

.89547 

8.56677 

60 

267 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


/ 

84° 

85° 

86° 

' 

Vers. 

Exsec. 

Yers. 

Exsec. 

Vers. 

Exsec. 

0 

.89547 

8.5G677 

.91284 

10.47371 

.93024 

13.33559 

0 

1 

.89576 

8.59332 

.91313 

10.51199 

.93053 

13.39547 

1 

2 

.89605 

8.62002 

.91342 

10.55052 

.93082 

13.45586 

2 

3 

.89634 

8.64687 

.91371 

10.58932 

.93111 

13.51676 

3 

4 

.89663 

8.67387 

.91400 

10.62837 

.93140 

13.57817 

4 

5 

.89092 

8.70103 

.91429 

10.66769 

.93169 

13.64011 

5 

6 

.  89721 

8.72833 

.91458 

10.70728 

.93198 

13.70258 

6 

7 

.89750 

8.75579 

.91487 

10.74714 

.93227 

13.76558 

7 

8 

.89779 

8.78341 

.91516 

10.78727 

.93257 

13.82913 

8 

9 

.89808 

8.81119 

.91545 

10.82768 

.93286 

13.80323 

9 

10 

.89836 

8.83912 

.91574 

10.86837 

.93315 

13.95788 

10 

11 

.89865 

8.86722 

.91603 

10.90934 

.93344 

14.02310 

11 

12 

.89894 

8.89547 

.91632 

10.95060 

.93373 

14.08890 

12 

13 

.89923 

8.92389 

.91661 

10.99214 

.93402 

14.15527 

13 

14 

.89952 

8.95248 

.91690 

11.03397 

.93431 

14.22223 

14 

15 

.89981 

8.98123 

.91719 

11.07610 

.93460 

14.28979 

15 

16 

.90010 

9.01015 

.91748 

11.11852 

.93489 

14.35795 

16 

17 

.90039 

9.03923 

.91777 

11.16125 

.93518 

14.42672 

17 

18 

.90088 

9.06849 

.91806 

11.20427 

.93547 

14.49611 

18 

19 

.90097 

9.09792 

.91835 

11.24761 

.93576 

14.56614 

19 

20 

.90126 

9.12752 

.91864 

11.29125 

.93605 

14.63679 

20 

21 

.90155 

9.15730 

.91893 

11.33521 

.93634 

14.70810 

21 

22 

.90184 

9.18725 

.91922 

11.37948 

.93663 

14.78005 

22 

23 

.90213 

9.21739 

.91951 

11.42408 

.93692 

14.85268 

23 

24 

.90242 

9.24770 

.91980 

11.46900 

.93721 

14.92597 

24 

25 

.90271 

9.27819 

.92009 

11.51424 

.93750 

14.99995 

25 

26 

.90300 

9.30887 

.92038  ' 

11.55982 

.93779 

15.07462 

26 

27 

.90329 

9.33973 

.92067 

11.60572 

.93808 

15.14999 

27 

28 

.90358 

9.87077 

.92096 

11.65197 

.93837 

15.22607 

28 

29 

.90386 

9.40201 

.92125 

11.69856 

.93866 

15.30287 

29 

30 

.90415 

9.43343 

.92154 

11.74550 

.93895 

15.38041 

30 

31 

.90444 

9.46505 

.92183 

11.79278 

.93924 

15.45869 

31 

32 

.90473 

9.49685 

'.92212 

11.84042 

.93953 

15.53772 

32 

33 

.90502 

9.52886 

.92241 

11.88841 

.93982 

15.61751 

33 

34 

.90531 

9.56106 

.92270 

11.93677 

.94011 

15.69808 

34 

35 

.90560 

9.59346 

.92299 

11.98549 

.94040 

15.77044 

35 

36 

.90589 

9.62605 

.92328 

12.03458 

.94069 

15.86159 

36 

37 

.90618 

9.65885 

.92357 

12.08040 

.94098 

15.94456 

37 

38 

.90647 

9.69186 

.92386 

12.13388 

.94127 

16.02835 

38 

39 

.90676 

9.72507 

.92415 

32.18411 

.94156 

16.11297 

39 

40 

.90705 

9.75849 

.92444 

12.23472 

.94186 

16.19843 

40 

41 

.90734 

9.79212 

.92473 

12.28572 

.94215 

16.28476 

41 

42 

.90763 

9.82596 

.92502 

12.33:12 

.94244 

16.37196 

43 

43 

.90792 

9.86001 

.92531 

12.38891 

.94273 

16.46005 

43 

44 

.90821 

9.89428 

.92560 

12.44112 

.94302 

16.54903 

41 

45 

.90850 

9.92877 

.92589 

12.49373 

.94331 

16.63893 

45 

46 

.90879 

9.96348 

.92618 

12.54676 

.94360 

16.72975 

46 

47 

.90908 

9.99841 

.92647 

12.60021 

.94389 

16.82152 

47 

48 

.90937 

10.03356 

.92676 

12.65408 

.94418 

16.91424 

48 

49 

.90966 

10.06894 

.92705 

12.70S38 

.94147 

17.C0794 

49 

50 

.90995  ' 

10.10455 

.92734 

12.70312 

.94476 

17.10262 

50 

51 

.91024 

10.14039 

.92763 

12.81829 

.94505 

17.19830 

51 

52 

.91053 

10.17646 

.92?'92 

12.87391 

.94534 

17.29501 

53 

53 

.91082 

10.21277 

.92821 

12.92999 

.94563 

17.39274 

53 

54 

.91111 

10.24932 

.92850 

12.98651 

.94592 

17.49153 

54 

55 

.91140 

10.28610 

.92879 

13  04350 

.94621 

17.59139 

55 

56 

.91169 

10.32313 

.92908 

13.10096 

.94650 

17.69233 

56 

57 

.91197 

10.36040 

.92937 

13.15889 

.94679 

17.79438 

57 

58 

.91226 

10.39792 

.92966 

13.21730 

.94708 

17.89755 

58 

59 

.91255 

10.43569 

.92995 

13.27620 

.94737 

18.00185 

59 

60 

.91284 

10.47371 

.93024 

13.33559 

.94766 

18.10732 

60 

268 


TABLE  XIII.— VERSINES  AND  EXSECANTS. 


/ 

87  3 

88° 

89° 

/ 

Vers. 

Exsec. 

Vers. 

Exsec. 

Vers. 

Exsec. 

0 

.94766 

18.10732 

.96510 

27.65371 

.98255 

56.29869 

0 

1 

.94795 

18.21397 

.96539 

27.89440  j 

.98284 

57.26976 

1 

.94825 

18,32182 

.96568 

28.13917 

.98313 

58.27431 

2 

3 

.94854 

18.43088 

.96597 

28.38812 

.98342 

59.31411 

3 

4 

.94833 

18.54119 

.96626 

28.64137 

.98371 

60.39105 

4 

5 

.94912 

18.65275 

.96655 

28.89903 

.98400 

61.50715 

5 

6 

.94941 

18.76560 

.96684 

29.16120 

.98429 

62.66460 

6 

7 

.94970 

18.87976 

.96714 

29.42802 

.98458 

63.86572 

7 

8 

.94999 

18.99524 

.96743 

29.69960 

.98487 

65.11304 

8 

9 

.95028 

19.11208 

.96772 

29.97607 

.98517 

66.40927 

9 

10 

.95057 

19.23028 

.96801 

30.25758 

.98546 

67.75736 

10 

11 

.95086 

19.34989 

.96830 

30.54425 

.98575 

69.16047 

11 

12 

.95115 

19.47093 

.96859 

30.83623 

.98604 

70.62285 

12 

13 

.95144 

19.59341 

.96888 

31.13366 

.98633 

72.14583 

13 

14 

.95173 

19.71737 

.96917 

31.43671 

.98662 

73.73586 

14 

15 

.95202 

19.84283 

.96946 

31.74554 

.98691 

75.39655 

15 

16 

.95231 

19.96982 

.96975 

32.06030 

.98720 

77.13274 

16 

17 

.95260 

20.09838 

.97004 

32.38118 

.98749 

78.94968 

17 

13 

.95289 

20.22852 

.97033 

32.70835 

.98778 

80.85315 

18 

19 

.95318 

20.36027 

.97062 

33.04199 

.98807 

82.84947 

19 

20 

.95347 

20.49368 

.97092 

33.38232 

.98836 

84.94561 

20 

21 

.95377 

20.62876 

.97121 

33.72952 

,98866 

87.14924 

21 

22 

.95406 

20.76555 

.97150 

34.08380 

.98895 

89.46886 

22 

23 

.95435 

20.90409 

.97179 

34.44539 

.98924 

91.91387 

23 

24 

.95464 

21.04440 

.97208 

34.81452 

.98953 

94.49471 

24 

25 

.95493 

21.18653 

.97237 

35.19141 

.98982 

97.22303 

25 

26 

.95522 

21.33050 

.97266 

35.57633 

.99011 

100.1119 

26 

27 

.95551 

21.47635 

.97295 

35.96953 

.99040 

103.1757 

27 

28 

.95580 

21.62413 

.97324 

36.37127 

.99069 

106.4311 

28 

29 

.95609 

21.77386 

.97353 

36.78185 

.1)9098 

109.8966 

29 

30 

.95638 

21.92559 

.97382 

37.20155 

.99127 

113.5930 

30 

31 

.95667 

22.07935 

.97411 

37.63068 

.09156 

117.5444 

31 

32 

.95696 

22.23520 

.97440 

38.06957 

.99186 

121.7780 

32 

33 

.95725 

22.39316 

.97470 

38.51855 

.99215 

126.3253 

33 

34 

.95754T 

22.55329 

.97499 

38.97797 

.99244 

131.2223 

34 

35 

.95783 

22.71563 

.97528 

39.44820 

.99273 

136.5111 

35 

36 

.95812 

22.88022 

.97557 

39.92963 

.99S02 

142.2406 

36 

37 

.95842 

23.04712 

.97586 

40.42266 

.99331 

148.4684 

37 

38 

.95871 

23.21637 

.97615 

40.92772 

.99360 

155.2623 

38 

39 

.95900 

23.38802 

.97644 

41.44525 

.99889 

162.7033 

39 

40 

.95929 

23.56212 

.97673 

41.97571 

.99418 

170.8883 

40 

41 

.95958 

23.73873 

.97702 

42.51961 

.99447 

179.9350 

41 

42 

.95987 

23.91790 

.97731 

43.07746 

.99476 

189.9868 

42 

43 

.96016 

24.09969 

.97760 

43.64980 

.99505 

201.2212 

43 

44 

.96045 

24.28414 

.97789 

44.23720 

.99535 

213.8600 

44 

45 

.96074 

24.47134 

.97819 

44.84026 

.99564 

228.1839 

45 

4G 

.96103 

24.66132 

.97848 

45.45963 

.99593 

244.5540 

46 

47 

.96132 

24.85417 

.97877 

46.09596 

.99622 

263.4427 

47 

48 

.96161 

25.04994 

.97906 

46.74997 

.99651 

285.4795 

48 

49 

.96190 

25.24869 

.97935 

47.42241 

.99C80 

311.5230 

49 

50 

.96219 

25.45051 

.97964 

48.11406 

.997'09 

342.7752 

50 

51 

.96248 

25.65546 

.97993 

48.82576 

.99738 

380.9723 

51 

52 

.90277 

25.86360 

.98022 

49.55840 

.99767 

428.7187 

52 

53 

.96307 

26.07503 

.98051 

50.31290 

.OC796 

490.1070 

53 

54 

.96336 

26.28981 

.98080 

51.09027 

.99825 

571.9581 

54 

55 

.96365 

26.50804 

.98109 

51.89156 

.99855 

686.5496 

55 

56 

.96394 

26.72978 

.98138 

52.71790 

.99884 

858.4369 

56 

57 

.96423 

26.95513 

.98168 

53.57046 

.99913 

1144.916 

57 

58 

.96452 

27.18417 

.98197 

64.45053 

.99942 

1717.874 

58 

59 

.96481 

27.41700 

.98226 

55.a5946 

.99971 

3436.747 

59 

60 

.96510 

27.65371 

.98255 

56.29869 

1.00000 

Infinite 

60 

269 


TABLE  XIV.-CUBIC  YARDS  PER  100  FEET.      SLOPES 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
22 

Base 
24 

Base 
26 

Base 
28 

1 

45 

53 

60 

68 

82 

90 

97 

105 

2 

93 

107 

122 

137 

167 

181 

196 

211 

3 

142 

163 

186 

208 

253 

275 

297 

319 

4 

193 

222 

252 

281 

341 

870 

400 

430 

5 

245 

282 

319 

356 

431 

468 

505 

542 

6 

300 

844 

389 

433 

522 

567 

611 

656 

7 

356 

408 

460 

512 

616 

668 

719 

771 

8 

415 

474 

533 

593 

711 

770 

830 

889 

9 

475 

542 

608 

675 

808 

875 

942 

1008 

10 

537 

611 

685 

759 

907 

981 

1056 

1130 

11 

601 

682 

764 

845 

1008 

1090 

1171 

1253 

12 

667 

756 

844 

933 

1111 

1200 

1289 

1378 

13 

734 

831 

926 

1023 

1216 

1312 

1408 

1505 

14 

804 

907 

1010 

1115 

1322 

1426 

1530 

1633 

15 

875 

986 

1096 

1208 

1431 

1542 

1653 

1764 

16 

948 

1067 

1184 

1304 

1541 

1659 

1778 

1896 

17 

1023 

1149 

1274 

1401 

1653 

1779 

1905 

2031 

18 

1100 

1233 

1366 

1500 

1767 

1900 

2033 

2167 

19 

1179 

1319 

1460 

1601 

1882 

2023 

2164 

2305 

20 

1259 

1407 

1555 

1704 

2000 

2148 

2296 

2444 

21 

1342 

1497 

1653 

1808 

2119 

2275 

2431 

2586 

22 

1426 

1589 

1752 

1915 

2241 

2404 

-  2567 

2730 

23 

1512 

1682 

1853 

2023 

2364 

2534 

2705 

2875 

24 

1600 

1778 

1955 

2133 

2489 

2667 

2844 

8022 

25 

1690 

1875 

2060 

2245 

2616 

2801 

2986 

3171 

26 

1781 

1974 

2166 

2359 

2744 

2937 

3130 

3322 

27 

1875 

2075 

2274 

2475 

2875 

3075 

3275 

3475 

28 

1970 

2178 

2384 

2593 

3007 

3215 

3422 

3630 

29 

2068 

2282 

2496 

2712 

3142 

3358 

3571 

3786 

30 

2167 

2389 

2610 

2833 

3278 

8500 

3722 

3944 

31 

2268 

2497 

2726 

2956 

3416 

8645 

3875 

4105 

32 

2370 

2607 

2844 

3081 

3556 

8793 

4030 

4267 

33 

2475 

2719 

2964 

3208 

3697 

3942 

4186 

4431 

34 

2581 

2833 

3085 

3337 

3841 

4093 

4344 

4596 

85 

2690 

2949 

3208 

3468 

3986 

4245 

4505 

4764 

36 

2800 

3067 

3333 

3600 

4133 

4400 

4667 

4933 

37 

2912 

3186 

8460 

8734 

4282 

4556 

4831 

5105 

88 

3026 

3307 

3589 

3870 

4433 

4715 

4996 

5278 

39 

3142 

3431 

3719 

4008 

4586 

4875 

5164 

5453 

40 

3259 

3556 

3852 

4148 

4741 

5037 

5333 

5630 

41 

8379 

3682 

3986 

4290 

4897 

5201 

5505 

5808 

42 

3500 

3811 

4122 

4433 

5056 

5367 

5678 

5989 

43 

3623 

3942 

4260 

4579 

5216 

5534 

5853 

6171 

44 

3748 

4074 

4400 

4726 

5378 

5704 

6030 

6356 

45 

3875 

4208 

4541 

4875 

5542 

5875 

6208 

6542 

46 

4004 

4344 

4684 

5026 

5707 

6048 

6389 

6730 

47 

4134 

4482 

4830 

5179 

5875 

6223 

6571 

6919 

48 

4267 

4622 

4978 

5333 

6044 

6400 

6756 

7111 

49 

4401 

4764 

5127 

5490 

6216 

6579 

6943 

7305 

50 

4537 

4907 

5278 

5648 

6389 

6759 

7130 

7500 

51 

4675 

5053 

5430 

5808 

6564 

6942 

7319 

7697 

52 

4815 

5200 

5584 

5970 

6741 

7126 

7511 

7896 

53 

4956 

5349 

5741 

6134 

6919 

7312 

7705 

8097 

54 

5100 

5500 

5900 

6300 

7100 

7500 

7900 

8300 

55 

5245 

5653 

6060 

6468 

7282 

7690 

8097 

8505 

56 

5393 

5807 

6222 

6637 

7467 

7881 

8296 

8711 

57 

5542 

5964 

6386 

6808 

7653 

8075 

8497 

8919 

58 

5693 

6122 

6552 

6981 

7841 

8270 

8700 

9130 

59 

5845 

6282 

6719 

7156 

8031 

8468 

8905 

9342 

60 

6000 

6444 

6889 

7333 

8222 

8667 

9111 

9556 

270 


TABLE  XIV.  —CUBIC  YARDS  PER  100  FEET.      SLOPES 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
22 

Base 
24 

Base 
26 

Base 
28 

1 

46 

54 

61 

69 

83 

91 

98 

106 

2 

96 

111 

126 

141 

170 

185 

200 

215 

3 

150 

172 

194 

217 

201 

283 

306 

328 

4 

207 

237 

267 

296 

356 

385 

415 

444 

5 

269 

306 

343 

380 

454 

491 

528 

565 

6 

333 

378 

422 

467 

556 

600 

644 

689 

7 

402 

454 

506 

557 

661 

713 

765 

817 

8 

474 

533 

593 

652 

770 

830 

889 

948 

9 

550 

617 

683 

750 

883 

950 

1017 

1083 

10 

630 

'  704 

778 

852 

1000 

1074 

1148 

1222 

11 

713 

794 

876 

957 

1120 

1202 

1283 

1365 

13 

800 

889 

978 

1067 

1244 

1333 

1422 

1511 

13 

891 

987 

1083 

1180 

1372 

1469 

1565 

16G1 

985 

1089 

1193 

1296 

1504 

1607 

1711 

1815 

15 

1083 

1194 

1306 

1417 

1639 

1750 

1861 

1972 

16 

1185 

1304 

1422 

1541 

1779 

1896 

2015 

2133 

17 

1291 

1417 

1543 

1669 

1920 

2046 

2172 

2298 

13 

1400 

1533 

1667 

1800 

2067 

2200 

2333 

2467 

19 

1513 

1G54 

1794 

1935 

2217 

2357 

2498 

2639 

23 

1630 

1778 

1926 

2074 

2370 

2519 

2667 

•  2815 

21 

1750 

1906 

2061 

2217 

2528 

2683 

2839 

2994 

22 

1874 

2037 

2200 

2363 

2689 

2852 

3015 

3178 

23 

2002 

2172 

2343 

2513 

2854 

3024 

3194 

•8365 

24 

2133 

2311 

2489 

2667 

3022 

3200 

3378 

3556 

23 

2269 

2454 

2639 

2824 

3194 

3380 

35G5 

8750 

23 

2407 

2GOO 

2793 

2985 

3370 

&5C3 

3756 

3948 

27 

2550 

2750 

2950 

3150 

3550 

3750 

3950 

4151 

28 

2096 

2904 

3111 

3319 

8733 

3941 

4148 

4356 

29 

28-16 

3061 

3276 

3491 

3920 

4135 

4350 

45G5 

30 

3000 

3222 

3444 

3667 

4111 

4333 

4556 

4778 

81 

8157 

3387 

3617 

3846 

4306 

4535 

4765 

4994 

32 

3319 

3556 

3793 

4030 

4504 

4741 

4978 

5215 

33 

3483 

3728 

3972 

4217 

4706 

4950 

5194 

5439 

84 

3652 

3904 

4156 

4407 

4911 

5163 

5415 

5667 

35 

8824 

4083 

4343 

4602 

5120 

5380 

5639 

5898 

36 

4000 

4267 

4533 

4800 

5333 

5600 

5867 

6133 

37 

4180 

4454 

4728 

5002 

5550 

5824 

6098 

6372 

38 

4363 

4644 

4926 

5207 

5770 

6052 

6333 

6615 

39 

4550 

4839 

5128 

5417 

5994 

6283 

6572 

6861 

40 

4741 

5037 

5333 

5630 

6222 

6519 

6815 

7111 

41 

4935 

5239 

5543 

5846 

6454 

6757 

7061 

7365 

42 

5133 

5444 

5756 

6067 

6689 

7000 

7311 

7623 

43 

5335 

5654 

5072 

6291 

6928 

7246 

7565 

7883 

44 

5541 

5867 

6193 

6519 

7170 

7496 

7822 

8148 

45 

5750 

6083 

6417 

6750 

7417 

7750 

8083 

8417 

46 

5963 

6304 

6644 

6985 

7667 

8007 

8348 

8689 

47 

6180 

6528 

6876 

7224 

7920 

8269 

8617 

8965 

41 

6400 
6624 

6756 
6987 

7111 

7350 

.7467 
7713 

8178 
8439 

m 

9244 
9528 

50 

6852 

7222 

7593 

7963 

87C4 

9074 

9444 

9815 

51 

7083 

7461 

7839 

8217 

8972 

9350 

9728 

10106 

52 

7319 

7704 

8089 

8474 

9244 

9G30 

10015 

10400 

53 

7557 

7950 

8343 

8735 

9520 

9913 

10306 

10698 

54 

7800 

8200 

8600 

9000 

9800 

10200 

10600 

11000 

55 

8046 

8454 

8861 

9269 

10083 

10491 

10898 

11306 

56 

8296 

8711 

9126 

9541 

10370 

10785 

11200 

11615 

57 

8550 

8972 

9394 

9817 

10661 

11083 

11506 

11928 

58 

8807 

9237 

9667 

10096 

10956 

11385 

11815 

12244 

59 

9069 

9506 

9943 

10380 

11254 

11691 

12128 

12565 

60 

9333 

9778 

10222 

10667 

11556 

12000 

12444 

12889 

271 


TABLE  XIV.— CUBIC  YARDS  PER  100  FEET.      SLOPES   1  •  1. 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
20 

Base 
28 

Base 
30 

Base 
32 

1 

48 

S6 

63 

70 

78 

107 

115 

122 

2 

104 

119 

133 

148 

163 

222 

237 

252 

3 

167 

189 

211 

233 

256 

344 

367 

389 

4 

237 

267 

296 

326 

356 

474 

504 

533 

5 

315 

352 

389 

426 

463 

611 

648 

685 

C 

400 

444 

489 

533 

578 

756 

800 

844 

7 

493 

544 

596 

648 

700 

907 

959 

1011 

8 

593 

652 

711 

770 

830 

1067 

1126 

1185 

9 

700 

767 

833 

900 

967 

1233 

1300 

13G7 

10 

815 

889 

963 

1037 

1111 

1407 

1481 

1556 

11 

937 

1019 

1100 

1181 

1263 

1589 

1670 

1752 

12 

10G7 

1156 

1244 

1333 

1422 

1778 

1867 

1956 

13 

1204 

1300 

1396 

1493 

1589 

1974 

2070 

2167 

14 

1348 

1452 

1556 

1659 

1763 

2178 

2281 

2385 

15 

1500 

1611 

1722 

1833 

1944 

2389 

2500 

2611 

16 

1659 

1778 

1896 

2015 

2133 

2607 

2726 

2844 

17 

1826 

1952 

2078 

2204 

2330 

2833 

2959 

3085 

18 

2000 

2133 

2267 

2400 

2533 

3067 

3200 

3333 

19 

2181 

2322 

2463 

2604 

2744 

3307 

3418 

35G9 

20 

2370 

2519 

2667 

2815 

2963 

3556 

3704 

3852 

21 

2567 

2722 

2878 

3033 

3189 

3811 

3967 

4122 

2770 

2933 

3096 

3259 

3422 

4074 

4237 

4144 

23 

2981 

3152 

3322 

3193 

3663 

4344 

4515 

46S5 

24 

3200 

3378 

3556 

3733 

3911 

4622 

4800 

4978 

25 

3426 

3611 

3796 

3981 

4167 

4907 

5093 

5278 

26 

3659 

3852 

4044 

4237 

4430 

5200 

5393 

5585 

27 

3900 

4100 

4300 

4500 

4700 

5500 

5700 

5900 

28 

4148 

4356 

4563 

4770 

4978 

5807 

6015 

6222 

29 

4404 

4619 

4833 

5048 

5263 

G122 

6337 

6552 

30 

4667 

4889 

5111 

5333 

5556 

6444 

6667 

6889 

31 

4937 

5167 

5396 

5626 

5856 

6774 

7004 

7233 

32 

5215 

5452 

5689 

5926 

6163 

7111 

7348 

7585 

33 

5500 

5744 

5989 

6233 

6478 

7456 

7700 

7944 

84 

5793 

6044 

6296 

6548 

6800 

7807 

8059 

8311 

35 

6093 

6352 

6611 

6870 

7130 

8167 

8426 

8685 

36 

6400 

6667 

6933 

7200 

7467 

8533 

8800 

90C7 

37 

6715 

6989 

7263 

7537 

7811 

8907 

9181 

94C6 

38 

7037 

7319 

7600 

7881 

8163 

9289 

9570 

9852 

39 

7367 

7656 

7944 

8233 

8522 

9678 

9967 

10256 

40 

7704 

8000 

8296 

8593 

8889 

10074 

10370 

106G7 

41 

8048 

8352 

8656 

8959 

9263 

10478 

10781 

11085 

42 

8400 

8711 

9022 

9333 

9644 

10889 

11200 

11511 

43 

8759 

9078 

9396 

9715 

10033 

11307 

11626 

11944 

44 

9126 

9452 

9778 

10104 

10430 

11733 

12059 

12385 

45 

9500 

9833 

10167 

10500 

10833 

12167 

12500 

12833 

46 

9881 

10222 

10563 

10904 

11244 

12607 

12948 

13289 

47 

10270 

10619 

10367 

11315 

11663 

13056 

13404 

13752 

48 

10667 

11022 

11378 

11733 

12089 

13511 

13867 

14222 

49 

11070 

114:33 

11796 

12159 

12522 

13974 

14337 

14700 

50 

11481 

11852 

12222 

12593 

12963 

14444 

14815 

15185 

51 

11900 

12278 

12656 

13033 

13411 

14922 

15300 

15678 

52 

12326 

12711 

13096 

13481 

13867 

15407 

15793 

16173 

53 

12759 

13152 

13544 

13937 

14330 

15900 

16293 

16685 

54 

13200 

13600 

14000 

14400 

14800 

16400 

16800 

17200 

55 

13648 

14056 

14463 

14870 

15278 

16907 

17315 

17722 

56 

14104 

14519 

14933 

15348 

15763 

17422 

17837 

18252 

67 

14567 

14989 

15411 

15833 

16256 

17944 

18367 

18789 

58 

15037 

15467 

15896 

16326 

16756 

18474 

18904 

19333 

59 

15515 

15952 

16389 

16826 

17263 

19011 

19448 

198R5 

60 

16000 

16444 

16889 

17333 

17778 

19556 

20000 

20444 

272 


TABLE  XIV,— CUBIC  YARDS  PER  100  FEET.     SLOPES  l^J  1  L 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
20 

Base 
28 

Base 
30 

Base 
32 

1 

50 

57 

65 

72 

80 

109 

117 

124 

2 

111 

126 

141 

156 

170 

230 

244 

259 

3 

183 

206 

228 

250 

272 

361 

383 

406 

4 

267 

296 

326 

356 

385 

504 

533 

5G3 

5 

361 

398 

435 

472 

509 

657 

694 

731 

6 

467 

511 

556 

000 

644 

822 

867 

911 

7 

583 

635 

687 

739 

791 

998 

1050 

1102 

8 

711 

770 

830 

889 

948 

1185 

1244 

1304 

9 

850 

917 

983 

1050 

1116 

1383 

1450 

1517 

10 

1000 

1074 

1148 

1222 

1296 

1593 

1667 

1741 

11 

1161 

1243 

1324 

1406 

1487 

1813 

1894 

1976 

12 

1333 

1422 

1511 

1600 

1689 

2044 

2133 

2222 

13 

1517 

1613 

1709 

1806 

1902 

2287 

2383 

2480 

14 

1711 

1815 

1919 

2022 

2126 

2541 

2644 

2743 

15 

1917 

2028 

2139 

2250 

2361 

2808 

2917 

3028 

16 

2133 

2252 

2370 

2489 

2607 

3081 

3200 

3319 

17 

2361 

2487 

2613 

2739 

2865 

33G9 

3494 

•90 

18 

2600 

2733 

2867 

3000 

3133 

36G7 

3800 

3933 

19 

2850 

2991 

3131 

3272 

3413 

3976 

4117 

4257 

20 

3111 

3259 

3407 

3556 

3704 

4296 

4444 

4592 

21 

3383 

3539 

3694 

3850 

4005 

4628 

4783 

4939 

22 

3667 

3830 

3993 

4156 

4318 

4970 

5133 

5296 

23 

3961 

4131 

4302 

4472 

4642 

5324 

5494 

56G5 

24 

4267 

4444 

4622 

4800 

4978 

5689 

58G7 

6044 

25 

4583 

4769 

4954 

5139 

5324 

60G5 

6250 

6435 

26 

4911 

5104 

5296 

5489 

5681 

6452 

6644 

6837 

27 

5250 

5450 

5650 

5850 

6050 

6850 

7050 

7250 

28 

5600 

5807 

6015 

6222 

6430 

7259 

7467 

7674 

29 

5961 

6176 

6391 

6606 

6820 

7680 

7894 

8109 

30 

6333 

6556 

6778 

7000 

7222 

8111 

8333 

8555 

31 

6717 

6946 

7176 

7406 

7635 

8554 

8783 

9013 

32 

7111 

7348 

7585 

7822 

8059 

9007 

9244 

9482 

33 

7517 

7761 

8006 

8250 

8494 

9472 

9717 

9962 

34 

7933 

8185 

8437 

86S9 

8941 

9948 

10200 

10452 

35 

8361 

8620 

8880 

9139 

9398 

10435 

10694 

10954 

36 

8800 

9067 

9333 

9600 

9867 

10933 

11200 

114G7 

37 

9250 

9524 

9798 

10072 

10346 

11443 

11717 

11991 

38 

9711 

9993 

10274 

10556 

10837 

11963 

12244 

125?S 

39 

10183 

10472 

10761 

11050 

11339 

12494 

12783 

13072 

40 

10667 

10963 

11259 

11556 

11852 

13037 

13333 

13G30 

41 

11161 

11465 

11769 

12072 

12376 

13591 

13894 

14198 

42 

11667 

11978 

12289 

12GOO 

12911 

14156 

14467 

14778 

43 

12183 

12502 

12820 

13139 

13457 

14731 

15050 

15369 

44 

12711 

13037 

13363 

13689 

14015 

15319 

15644 

15970 

45 

13250 

13583 

13917 

14250 

14583 

15917 

16250 

16583 

46 

1:3800 

14141 

14481 

14822 

15163 

16526 

16867 

17207 

47 

14361 

14709 

15057 

15406 

15754 

17146 

17494 

17843 

48 

14933 

15289 

15644 

16000 

16356 

17778 

18133 

18489 

49 

15517 

15880 

16243 

16606 

16968 

18420 

18783 

19146 

50 

16111 

16481 

16852 

17222 

17592 

19074 

19444 

19815 

51 

16717 

17094 

17472 

17850 

18228 

19739 

20117 

20494 

52 

17333 

17719 

18104 

18489 

18874 

20415 

20800 

21185 

53 

17961 

18354 

18746 

19139 

19531 

21102 

21494 

21887 

54 

18000 

19000 

19400 

19800 

20200 

21800 

22200 

22600 

55 

10250 

19657 

20065 

20472 

20880 

22509 

22917 

23324 

56 

19011 

20326 

20741 

21156 

21570 

23230 

23644 

24059 

•  57 

20583 

21006 

21428 

21850 

22272 

23961 

24383 

24805 

58 

21267 

21696 

22126 

22556 

22985 

24704 

251:33 

25563 

59 

21961 

22398 

2^835 

23272 

23709 

25457 

25894 

26332 

60 

22G07 

23111 

23556 

24000 

24444 

2G222 

26667 

27111 

273 


. 


TABLE   XTV. — CUBIC  YAUDS   PER   100  FEET.       SLOPES   2    ;  1. 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 

20 

Ba«e 

28 

Base 
30 

Base 
32 

1 

52 

59 

67 

74 

81 

ill 

119 

126 

2 

119 

133 

143 

1C3 

178 

237 

252 

267 

3 

200 

222 

244 

267 

289 

378 

400 

422 

4 

296 

326 

356 

385 

415 

533 

563 

693 

5 

407 

444 

481 

519 

556 

704 

741 

778 

6 

533 

578 

622 

6G7 

711 

889 

933 

978 

7 

674 

726 

778 

830 

881 

1089 

1141 

1193 

8 

830 

889 

948 

1007 

1067 

1304 

1363 

1422 

9 

1000 

1067 

1133 

1200 

1267 

1533 

1600 

1667 

10 

1185 

1259 

1333 

1407 

1481 

1778 

1852 

1926 

11 

1385 

1467 

1548 

1630 

1711 

2037 

2119 

2200 

12 

1600 

1689 

1778 

1867 

1956 

2311 

2400 

2489 

13 

1830 

1926 

2022 

2119 

2215 

2600 

2696 

2703 

14 

2074 

2178 

2281 

2385 

2489 

2904 

3007 

3111 

15 

2333 

2444 

2556 

2667 

2778 

3222 

3333 

3444 

16 

2607 

2726 

2844 

2963 

3081 

3556 

3674 

3793 

17 

2896 

3022 

3148 

3274 

3400 

3904 

4030 

4156 

18 

3200 

S333 

3467 

3600 

3733 

4267 

4400 

4533 

19 

3319 

3659 

3800 

3941 

4081 

4644 

4785 

4926 

20 

3852 

4000 

4148 

4296 

4444 

5037 

5185 

5333 

21 

4200 

4356 

4511 

4667 

4822 

5444 

5600 

5756 

22 

4563 

4730 

4889 

5052 

5215 

5867 

6030 

81D8 

23 

4941 

5111 

5281 

5452 

5622 

6304 

6474 

6644 

24 

5333 

5511 

5689 

5867 

6044 

6756 

6933 

7111 

25 

5741 

5926 

6111 

6296 

6481 

7522 

7407 

7593 

26 

6163 

6356 

6548 

6741 

6933 

7704 

7896 

KO!-'9 

27 

6600 

6800 

7000 

7200 

7400 

8200 

8400 

8600 

28 

7052 

7259 

7467 

7674 

7881 

8711 

8919 

9126 

29 

7519 

7733 

7948 

8163 

8378 

9237 

9452 

9ii67 

30 

8000 

8222 

8444 

8667 

8389 

9778 

10000 

10222 

31 

8496 

8729 

8956 

9185 

9415 

10333 

10563 

10793 

32 

9007 

9244 

9481 

9719 

9958 

10904 

11141 

11378 

33 

9533 

9778 

10022 

10267 

10511 

11489 

11733 

11978 

34 

10074 

10326 

10578 

10330 

11081 

12089 

1-2341 

12593 

35 

10830 

10889 

11148 

11407 

11G67 

1-2704 

12963 

13222 

36 

11200 

11467 

11733 

12000 

12267 

13333 

13600 

13867 

37 

11785 

12059 

12333 

12607 

12S81 

131)78 

14252 

14526 

38 

12385 

12667 

12948 

13230 

13511 

14ii37 

14919 

15200 

39 

13000 

13289 

13578 

13867 

14156 

15311 

15600 

15889 

40 

13630 

13926 

14222 

14519 

14815 

16000 

16296 

16593 

41 

14274 

14578 

14881 

15185 

15489 

16704 

17007 

17311 

42 

14:83 

15244 

155C6 

15867 

16178 

17.22 

17733 

18044 

43 

15607 

15926 

16224 

16o63 

16881 

18156 

18474 

18793 

44 

16296 

16622 

16948 

17274 

17600 

18904 

19230 

19556 

45 

irooo 

17333 

17C67 

18000 

18333 

19667 

20000 

20333 

46 

17719 

18059 

18400 

18741 

19081 

20444 

20785 

21126 

47 

13152 

18800 

19148 

19496 

19S44 

21237 

21585 

21933 

48 

19200 

19556 

19911 

20267 

20<>22 

22344 

22400 

22756 

49 

19963 

20326 

20689 

21052 

21415 

22867 

23230 

2355)3 

50 

20741 

20711 

21481 

21852 

22222 

23704 

24074 

24444 

51 

2U33 

21911 

22289 

22667 

23044 

2-;5'6 

24933 

25311 

52 

22.341 

22726 

23111 

23496 

23881 

25422 

25807 

26193 

53 

23163 

23556 

2-3948 

24341 

24733 

26304 

26696 

27089 

54 

24000 

24400 

24800 

25200 

2:>600 

27200 

27600 

28000 

55 

24852 

25259 

25667 

26074 

26481 

28111 

28519 

28926 

56 

25719 

26133 

26548 

26963 

27378 

29037 

29452 

29867 

57 

20600 

27022 

27444 

27867 

28289 

2i)978 

30400 

30822- 

58 

27496 

27926 

28356 

23785 

29215 

30933 

31363 

31793 

59 

28407 

28844 

29281 

29719 

30156 

31904 

32341 

32778 

CO 

89333 

29778 

30222 

30667 

31111 

32889 

33333 

33778 

274 


TABLE  XIV.— CUBIC  YARDS  PER  100  FEET.      SLOPES  3  :  I. 


Depth 

Base 
12 

Base 
14 

Base 
16 

Base 
18 

Base 
20 

Base 
28 

Base 
30 

Base 
32 

1 

56 

63 

70 

78 

85 

115 

122 

130 

2 

133 

148 

163 

178 

193 

252 

267 

281 

3 

233 

256 

278 

300 

322 

411 

433 

456 

4 

356 

385 

415 

444 

474 

593 

622 

652 

5 

500 

537 

574 

611 

648 

796 

833 

870 

6 

667 

711 

756 

800 

844 

1022 

1067 

1111 

7 

856 

907 

959 

1011 

1063 

1270 

1322 

1374 

8 

1067 

1126 

1185 

1244 

1304 

1541 

1600 

1659 

9 

1300 

1367 

1433 

1500 

1567 

1833 

1900 

1967 

10 

1556 

1630 

1704 

1778 

1852 

2148 

2222 

2296 

11 

1833 

1915 

1996 

2078 

2159 

2485 

2567 

2648 

13 

2133 

2222 

2311 

2400 

2489 

2844 

2933 

3022 

13 

2456 

2552 

2648 

2744 

2841 

3226 

3322 

3419 

14 

2800 

2904 

3007 

3111 

3215 

3630 

3733 

3837 

15 

3167 

3278 

3389 

3500 

3611 

4056 

4167 

4278 

16 

3556 

3674 

3793 

3911 

4030 

4504 

4622 

4741 

17 

3967 

4093 

4219 

4344 

4470 

4974 

5100 

5226 

18 

4400 

4533 

4667 

4800 

4933 

5467 

5600 

5733 

19 

4856 

4996 

5137 

5278 

5419 

5981 

6122 

6263 

20 

5333 

5481 

5630 

5778 

5926 

6519 

6667 

6815 

21 

5833 

5989 

6144 

6300 

6456 

7078 

7233 

7389 

22 

6356 

6519 

6681 

6844 

7007 

7659 

7822 

7985 

23 

6900 

7070 

7241 

7411 

7581 

8263 

8433 

8504 

24 

7467 

7644 

7822 

8000 

8178 

8889 

9067 

9141 

25 

8056 

8241 

8426 

8611 

8796 

9537 

9722 

9807 

26 

8667 

8859 

9052 

9244 

9437 

10207 

10400 

10593 

27 

9300 

9500 

9700 

9900 

10100 

10900 

11100 

11300 

28 

9956 

10163 

10370 

10578 

10785 

11615 

11822 

12030 

29 

10633 

10848 

11063 

11278 

11493 

12352 

12567 

12781 

30 

11333 

11556 

11778 

12000 

12222 

13111 

13333 

13556 

31 

12056 

12285 

12515 

12744 

12974 

13893 

14122 

14352 

32 

12800 

13037 

13274 

13511 

13748 

14696 

14933 

15170 

33 

13567 

13811 

14056 

14300 

14544 

15522 

15767 

16011 

34 

14356 

14607 

14859 

15111 

16370 

16622 

16874 

35 

15167 

15426 

15685 

15944 

16204 

17241 

17500 

17759 

36 

16000 

16267 

16533 

16800 

17067 

18133 

18400 

18667 

37 

16856 

17130 

17404 

17678 

17952 

19048 

19322 

19596 

38 

17733 

18015 

18296 

18578 

18859 

19985 

20267 

20548 

39 

18633 

18922 

19211 

19500 

19789 

20944 

21233 

21522 

40 

19556 

19852 

20148 

20444 

20741 

21926 

22222 

22516 

41 

20500 

20804 

21107 

21411 

21715 

22930 

23233 

23537 

42 

21467 

21778 

22089 

22400 

22711 

23956 

24267 

24578 

43 

22456 

22774 

23093 

23411 

23730 

25004 

25322 

25641 

44 

23467 

23793 

24119 

24444 

24770 

26074 

26400 

26726 

45 

24500 

24833 

25167 

25500 

25833 

27167 

27500 

27833 

46 

25556 

25896 

26237 

26578 

26919 

28281 

28622 

28963 

47 

26633 

26981 

27330 

27678 

28026 

29419 

29767 

80115 

48 
49 

IS 

28444 

29581 

m 

29156 
30307 

30578 
31759 

30933 
32122 

31289 
32485 

50 

30000 

30370 

30741 

31111 

31481 

32963 

33333 

33704 

51 

31167 

31544 

31922 

32300 

32678 

34189 

34567 

34944 

52 

32356 

32741 

33126 

33511 

33896 

35437 

35822 

36207 

53 

33567 

33959 

34352 

34744 

35137 

36707 

37100 

87493 

54 

34800 

35200 

05600 

36000 

36400 

38000 

38400 

88800 

55 

36056 

36463 

36870 

37278 

37685 

39315 

39722 

4C130 

56 

37333 

37748 

38163 

38578 

38993 

40652 

41067 

41481 

57 

38633 

39056 

39478 

39900 

40322 

42011 

42433 

42856 

58 

39956 

40385 

40815 

41244 

41674 

43393 

43822 

44252 

59 

41300 

41737 

42174 

42611 

43048 

44796 

45233 

45670 

60 

42667 

43111 

43556 

44000 

44444 

46222 

46667 

47111 

275 


TABLE  XV.— CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area. 
Sq. 
Ft. 

Cubic  , 
Yards. 

Area. 

I?: 

Cubic 
Yards. 

Area. 

Sq. 
Ft. 

Cubic 
Yards. 

Area. 
Sq. 
Ft. 

Cubic 
Yards. 

Area. 

& 

Cubic 
Yards. 

1 

3.7 

51 

188.9 

101 

374.1 

151 

559.3 

201 

744.4 

2 

7.4 

52 

192.6 

102 

377.8 

152 

563.0 

202 

748.2 

3 

11.1 

53 

196.3 

103 

381.5 

153 

566.7 

203 

751.9 

4 

14.8 

54 

200.0 

104 

385.2 

154 

570.4 

204 

755.6 

5 

18.5 

55 

203.7 

105 

388.9 

155 

574.1 

205 

759.3 

6 

22.2 

56 

207.4 

106 

392.6 

156 

577.8 

206 

763.0 

7 

25.9 

57 

211.1 

107 

396.3 

157 

581.5 

207 

766.7 

8 

29.6 

58 

214.8 

108 

400.0 

158 

585.2 

208 

770.4 

9 

33.3 

59 

218.5 

109 

403.7 

159 

588.9 

209 

774.1 

10 

37.0 

60 

222.2 

110 

407.4 

160 

592.6 

210 

777.8 

11 

40.7 

61 

225.9 

111 

411.1 

161 

596.3 

211 

781.5 

12 

44.4 

62 

229.6 

112 

414.8 

162 

600.0 

212 

785.2 

13 

48.1 

63 

233.3 

113 

418.5 

163 

603.7 

213 

788.9 

14 

51.9 

64 

237.0 

114 

422.2 

164 

607.4 

214 

792.6 

15 

55.6 

65 

240.7 

115 

425.9 

165 

611.1 

215 

796.3 

16 

59.3 

66 

244.4 

116 

429.6 

166 

614.8 

216 

800.0 

17 

63.0 

67 

248.2 

117 

433.3 

167 

618.5 

217 

803.7 

18 

66.7 

68 

251.9 

118 

437.0 

168 

622.2 

218 

807.4 

19 

70.4 

69 

255.6 

119 

440.7 

169 

625.9 

219 

811.1 

20 

74.1 

70 

259.3 

120 

444.4 

170 

629.6 

220 

814.8 

21 

77.8 

71 

263.0 

121 

448.2 

171 

633.3 

221 

818.5 

22 

81.5 

72 

266.7 

122 

451.9 

172 

637  0 

222 

822.2 

23 

85.2 

73 

270.4 

123 

455.6 

173 

640.7 

223 

825.9 

24 

88.9 

74 

274.1 

124 

459.3 

174 

644.4 

204 

829.6 

25 

92  6 

75 

277.8 

125 

403.0 

175 

648.2 

225 

833.3 

28 

96.3 

76 

281.5 

126 

466.7 

176 

651.9 

226 

837.0 

27 

100.0 

77 

285  2 

127 

470.4 

177 

655.6 

227 

840.7 

28 

103.7 

78 

288.9 

128 

474.1 

178 

659.3 

228 

844.4 

29 

107.4 

79 

292.6 

129 

477.8 

179 

663.0 

229 

848.2 

30 

111.1 

80 

296.3 

130 

481.5 

180 

666.7 

230 

851.9 

31 

114.8 

81 

300.0 

131 

485.2 

181 

670.4 

231 

855.6 

32 

us.  5 

82 

303.7 

132 

488.9 

182 

674.1 

232 

859.3 

33 

122.2 

83 

307.4 

133 

492.6 

183 

677.8 

233 

8G3.0 

34 

125.9 

84 

311.1 

134 

496.3 

184 

681.5 

234 

866.7 

35 

129  6 

85 

314.8 

135 

500.0 

185 

685.2 

235 

870.4 

36 

133.3 

86 

318.5 

136 

503.7 

186 

688.9 

236 

874.1 

37 

137.0 

87 

322.2 

137 

507.4 

187 

692.6 

237 

877.8 

38 

140.7 

88 

325.9 

138 

511.1 

188 

696.3 

238 

881.5 

39 

144  4 

89 

329.6 

139 

514.8 

189 

700.0 

239 

885.2 

40 

148.2 

90 

333.3 

140 

518.5 

190 

703.7 

240 

888.9 

41 

151.9 

91 

387.0 

141 

522.2 

191 

707.4 

241 

892.6 

42 

1.">.6 

92 

340.7 

142 

525.9 

192 

711.1 

242 

896.3 

43 

159.3 

93 

344.4 

143 

529.6 

193 

714.8 

243 

900.0 

44 

1G3.0 

94 

348.2 

144 

533.3 

194 

718.5 

244 

90.1.7 

45 

166.7 

95 

351.9 

145 

537.0 

195 

722.2 

245 

907.4 

46 

170.4 

96 

355.6 

146 

540.7 

196 

725.9 

246 

911.1 

47 

174.1 

97 

359.3 

147 

544.4 

197 

729.6 

247 

914.8 

48 

177.8 

98 

363.0 

148 

548.2 

198 

733.3 

248 

918.5 

49 

181.5 

99 

306.7 

149 

551.9 

199 

737.0 

249 

9-.2.2 

50 

185.2 

100 

370.4 

150 

555.6 

200 

740.7 

250 

925.9 

276 


TABLE  XV.— CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area. 

$ 

Cubic 
Yards. 

Area. 

ft 

Cubic 
Yards. 

Area. 

ft 

Cubic 
Yards. 

Area. 

ft 

Cubic 
Yards. 

Area. 

ft 

Cubic 
Yards. 

251 

929.6 

301 

1114.8 

351 

1300.0 

401 

1485.2 

451 

1670.4 

252 

933  3 

302 

1118.5 

352 

1303.7 

402 

1488.9 

452 

1674.1 

253 

937.0 

303 

1122.2 

353 

1307.4 

403 

1492.6 

453 

1677.8 

254 

940.7 

304 

1125.9 

354 

1311.1 

404 

1496.3 

454 

1681.5 

255 

944.4 

305 

1129.6 

355 

1314.8 

405 

1500.0 

455 

1685.2 

256 

948.2 

306 

1133.3 

356 

1318.5 

406 

1503.7 

456 

1688.9 

257 

951  9 

307 

1137.0 

357 

1322.2 

407 

1507.4 

457 

1692.6 

258 

955.6 

308 

1140.7 

358 

1325.9 

408 

1511.1 

458 

1696.3 

259 

959.3 

309 

1144.4 

359 

1329.6 

409 

1514.8 

459 

1700.0 

260 

963.0 

310 

1148.2 

360 

1333.3 

410 

1518.5 

460 

1703.7 

261 

966.7 

311 

1151.9 

361 

1337.0 

411 

1522.2 

461 

1707.4 

262 

970.4 

312 

1155.6 

362 

1340.7 

412 

1525.9 

462 

1711.1 

203 

974  1 

313 

1159.3 

363 

1344.4 

413 

1529.6 

463 

1714.8 

264 

977.8 

314 

1163.0 

364 

1348.2 

414 

1533.3 

464 

1718.5 

265 

981.5 

315 

1166.7 

365 

1351.9 

415 

1537.0 

465 

1722.2 

266 

985.2 

316 

1170.4 

366 

1355.6 

416 

1540.7 

466 

1725.9 

267 

988.9 

317 

1174.1 

367 

1359.3 

417 

1544.4 

467 

1729.6 

268 

992.6 

318 

1177.8 

368 

1363.0 

418 

1548.2 

468 

1733.3 

263 

996.3 

319 

1181.5 

369 

1366.7 

419 

1551.9 

469 

1737.0 

270 

1000.0 

320 

1185.2 

370 

1370.4 

420 

1555.6 

470 

1740.7 

271 

1003.7 

321 

1188.9 

371 

1374.1 

421 

1559.3 

471 

1744.4 

272 

1007.4 

322 

1192.6 

372 

1377.8 

422 

1563.0 

472 

1748.2 

273 

1011.  I 

323 

1196.3 

373 

1381.5 

423 

1566.7 

473 

1751.9 

274 

1014.8 

324 

1200.0 

374 

1385.2 

424 

1570.4 

474 

1755.6 

275 

1018.5 

325 

1203.7 

375 

138S.9 

425 

1574.1 

475 

1759.3 

276 

1022.2 

326 

1207.4 

376 

1392.6 

426 

1577.8 

476 

1763.0 

277 

1025.9 

327 

1211.1 

377 

1396.3 

427 

1581.5 

477 

1766.7 

278 

10-29.6 

328 

1214.8 

378 

1400.0 

428 

1585.2 

478 

1770.4 

279 

1033.3 

329 

1218.5 

379 

M03.7 

429 

1588.9 

479 

1774.1 

280 

1037.0 

330 

1222.2 

3SO 

1407.4 

430 

1592.6 

480 

1777.8 

281 

1040.7 

331 

1325.9 

381 

1411.1 

431 

1596.3 

481 

1781.5 

,'?82 

1044.4 

332 

1229.6 

382 

1414.8 

432 

1600.0 

482 

1785.2 

283 

1048.2 

333 

1233.3 

383 

1418.5 

433 

1603.7 

483 

1788.9 

284 

1051.9 

334 

1237.0 

384 

1422.2 

434 

1607.4 

484 

1792.6 

285 

1055.6 

335 

1240.7 

385 

1425.9 

435 

1611.1 

485 

1796.3 

286 

1059.3 

336 

1244.4 

386 

1429.6 

436 

1614.8 

486 

1800.0 

287 

1003.0 

337 

1248.2 

387 

1433.3 

437 

1618.5 

487 

1803.7 

288 

1066.7 

338 

1251.9 

388 

1437.0 

438 

1622.2 

488 

1807.4 

289 

1070.4 

339 

1255.6 

389 

1440.7 

439 

1625.9 

489 

1811.1 

290 

1074.1 

340 

1259.3 

390 

1444.4 

440 

1629.6 

490 

1814.8 

291 

1077.8' 

341 

1263.0 

391 

1448.2 

441 

1633.3 

491 

1818.5 

292 

1081.5 

342 

1266.7 

392 

1451.9 

442 

1637.0 

492 

1822.2 

293 

1085.2 

343 

1270.4 

393 

1455.6 

443 

1640.7 

493 

1825.9 

294 

1088.9 

344 

1274.1 

394 

1459.3 

444 

1644.4 

494 

1829.6 

295 

1092.6 

345 

1277.8 

395 

1463.0 

445 

1648.2 

495 

1833.3 

296 

1096.3 

346 

1281.5 

396 

1466.7 

446 

1651.9 

496 

1837.0 

297 

1100.0 

347 

1285.2 

397 

1470.4 

447 

1655.6 

497 

1840.7 

298 

1103.7 

348 

1288.9 

398 

1474.1 

448 

1659.3 

498 

1844.4 

299 

1107.4 

349 

1292.6 

399 

1477.8 

449 

1663.0 

499 

1848.2 

300 

1111.1 

350 

1296.3 

400 

1481.5 

450 

1666.7 

500 

1851.9 

277 


TABLE  XV.— CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area. 

ft 

Cubic 
Yards. 

Area. 

ft 

Cubic 
Yards. 

Area. 

ft 

Cubic 

Yards. 

Area. 

11: 

Cubic 
Yards. 

Area. 

ft 

Cubic 

Yards. 

501 

1855.6 

551 

2040.7 

601 

2225.9 

651 

2411.1 

701 

2596.3 

502 

1859.3 

552 

2044.4 

602 

2229.6 

652 

2414.8 

702 

2600.0 

503 

1863.0 

553 

2048.2 

603 

2233.3 

653 

2418.5 

703 

2603.7 

504 

1866.7 

554 

2051.9 

604 

2237.0 

654 

2422.2 

704 

2607.4 

505 

1870.4 

555 

2055.6 

605 

2240.7 

655 

2425.9 

705 

2611.1 

506 

1874.1 

556 

2059.3 

606 

2244.4 

656 

2429.6 

706 

2614.8 

507 

1877.8 

557 

2063.0 

607 

2248.2 

657 

24a3.3 

707 

2618.5 

508 

1881.5 

558 

2066.7 

608 

2251.9 

658 

2437.0 

708 

2622.2 

509 

1885.2 

559 

2070.4 

609 

2255.6 

659 

2440.7 

709 

2625.9 

510 

1888.9 

560 

2074.1 

610 

2259.3 

660 

2444.4 

710 

2629.6 

511 

1892.6 

561 

2077.8 

611 

2263.0 

661 

2448.2 

711 

2633.3 

512 

1896.3 

562 

2081.5 

612 

2266.7 

662 

2451.9 

712 

2637.0 

513 

1900.0 

563 

2085.2 

613 

2270.4 

663 

2455.6 

713 

2640.7 

514 

1903.7 

564 

2088.9 

614 

2274.1 

664 

2459.3 

714 

2644.4 

515 

1907.4 

565 

2092.6 

615 

2277.8 

665 

2463.0 

715 

2648.2 

516 

1911.1 

566 

2096.3 

616 

2281.5 

666 

2466.7 

716 

2651.9 

517 

1914.8 

567 

2100.0 

617 

2285.2 

667 

2470.4 

717 

2655.6 

518 

1918.5 

568 

2103.7 

618 

2288.9 

668 

2474.1 

718 

2659.3 

519 

1922.2 

569 

2107.4 

619 

2292.6 

669 

2477.8 

719 

2663.0 

520 

1925.9 

570 

2111.1 

620 

2296.3 

670 

2481.5 

720 

2666.7 

521 

1929.6 

571 

2114.8 

621 

2300.0 

671 

2485.2 

721 

2670.4 

523 

1933.3 

572 

2118.5 

622 

2303.7 

672 

2488.9 

722 

2674.1 

523 

1937.0 

573 

2122.2 

623 

2307.4 

673 

2492.6 

723 

2677.8 

524 

1940.7 

574 

2125.9 

624 

2311.1 

674 

2496.3 

724 

2681.5 

525 

1944.4 

575 

2129.6 

625 

2314.8 

675 

2500.0 

725 

2685.2 

526 

1948.2 

576 

2133.3 

626 

2318.5 

676 

2503.7 

726 

2688.9 

527 

1951.9 

577 

2137.0 

627 

2322.2 

677 

2507.4 

727 

2692.6 

528 

1955.6 

578 

2140.7 

628 

2325.9 

678 

2511.1 

728 

2696.3 

529 

1959.3 

579 

2144.4 

629 

2329.6 

679 

2514.8 

729 

2700.0 

530 

1963.0 

580 

2148.2 

630 

2333.3 

680 

2518.5 

730 

2703.7 

531 

1966.7 

581 

2151.9 

631 

2337.0 

681 

2522.2 

731 

2707.4 

532 

1970.4 

582 

2155.6 

632 

2340.7 

682 

2525.9 

732 

2711.1 

533 

1974.1 

583 

2159.3 

633 

2344.4 

683 

2529.6 

733 

2714.8 

534 

1977.8 

584 

2163.0 

634 

2348.2 

684 

2533.3 

734 

2718.5 

535 

1981.5 

585 

21G6.7 

635 

2351.9 

685 

2537.0 

735 

2722.2 

536 

1985.2 

586 

2170.4 

636 

2355.6 

686 

2540  7 

736 

2725.9 

537 

1988.9 

587 

2174.1 

637 

2359.3 

687 

2544.4 

737 

2729.6 

538 

1992.6 

588 

2177.8 

638 

2363  0 

688 

2548.2 

138 

2733.3 

539 

1996.3 

589 

2181.5 

639 

2366.7 

689 

2551.9 

739 

2737.0 

540 

2000.0 

590 

2185.2 

640 

2370.4 

690 

2555.6 

740 

2740.7 

541 

2003.7 

591 

2188.9 

641 

2374.1 

691 

2559.3 

741 

2744.4 

542 

2007.4 

592 

2192.6 

642 

2377.8 

692 

2563.0 

742 

2748.2 

543 

2011.1 

593 

2196.3 

643 

2381.5 

693 

2566.7 

743 

2751.9 

544 

2014.8 

594 

2200.0 

644 

2385.2 

694 

2570.4 

744 

2755.6 

545 

2018.5 

595 

2203.7 

645 

2388.9 

695 

2574.1 

745 

2759.3 

546 

2022.2 

596 

2207.4 

646 

2392.6 

696 

2577.8 

746 

2763.0 

547 

2025.9 

597 

2211.1 

647 

2396.3 

697 

2581.5 

747 

2766.7 

548 

2029.6 

598 

2214.8 

648 

2400.0 

698 

2585.2 

748 

2770.4 

549 

2033.3 

599 

2218.5 

649 

2403.7 

699 

2588.9 

749 

2774.1 

550 

2037.0 

600 

2222.2 

650 

2407.4 

700 

2592.6 

750 

2777.8 

278 


TABLE  XV.— CUBIC  YARDS  IN  100  FEET  LENGTH. 


Area. 

1?: 

Cubic 
Yards. 

Area. 
Sq. 
Ft. 

Cubic 
Yards. 

Area. 

a 

Cubic 

Yards. 

Area. 

11: 

Cubic 
Yards. 

Area. 

St 

Cubic 
Yards. 

751 

2781.5 

801 

2966.7 

851 

3151.9 

901 

3337.0 

951 

3522.2 

752 

2785.2 

802 

2970.4 

852 

3155.6 

902 

3340.7 

952 

3525.9 

753 

27'88.9 

803 

2974.1 

853 

3159.3 

903 

3344.4 

953 

3529.6 

754 

2792.6 

804 

2977.8 

854 

3163.0 

904 

3348.2 

954 

3533.3 

755 

2796.3 

805 

2981.  5 

855 

3166.7 

905 

3351.9 

955 

3537.0 

756 

2800.0 

806 

2985.2 

856 

3170.4 

906 

3355.6 

956 

3540.7 

757 

2803.7 

807 

2988.9 

857 

3174.1 

907 

3359.3 

957 

3544.4 

758 

2807.4 

808 

2992.6 

858 

3177.8 

908 

3363.0 

958 

3548.2 

7'59 

2811.1 

809 

2996.3 

859 

3181.5 

909 

3366.7 

959 

3551.9 

760 

2814.8 

810 

3COO.O 

860 

3185.2 

910 

3370.4 

960 

3555.6 

761 

2818.5 

811 

3003.7 

861 

3188.9 

911 

3374.1 

961 

3559.3 

762 

2822.2 

812 

3007.4 

862 

3192.6 

912 

3377.8 

962 

3563.0 

763 

2825.9 

813 

3011.1 

863 

3196.3 

913 

3381.5 

963 

3566.7 

764 

2829.6 

814 

3014.8 

864 

3200.0 

914 

3385.2 

964 

3570.4 

765 

2833  3 

815 

3018.5 

865 

3203.7 

915 

3488.9 

965 

3574.1 

766 

2837.0 

816 

3022.2 

866 

3207.4 

916 

3392.6 

966 

3577.8 

767 

2840.7 

817 

3025.9 

867 

3211.1 

917 

8396.8 

967 

3581.5 

768 

2844.4 

818 

3009.6 

868 

3214.8 

918 

3400.0 

968 

3585.2 

769 

2848.2 

819 

3033.3 

869 

3v!l8.5 

919 

3403.7 

969 

3588.9 

770 

2851.9 

820 

3037.0 

870 

3222.2 

920 

3407.4 

970 

3592.6 

771 

2855.6 

821 

3040.7 

871 

3225.9 

921 

3411.1 

971 

3596.3 

772 

2859.3 

822 

3044.4 

872 

3229.6 

922 

3414.8 

972 

3600.0 

773 

2863.0 

823 

3048.2 

873 

3233.3 

923 

3-118.5 

973 

3603.7 

774 

2866.7 

824 

3051.9 

874 

3237.0 

924 

3422.2 

974 

3607.4 

775 

2870.4 

825 

3055.6 

875 

3240.7 

925 

3425.9 

975 

3611.1 

776 

2874.1 

826 

3059.3 

876 

3244.4 

926 

3429.6 

976 

3614.8 

777 

2877.8 

827 

3063.0 

877 

3248.2 

927 

3433.3 

977 

3618.5 

778 

2881.5 

828 

3066.7 

87'8 

3251.9 

928 

3437.0 

978 

3622.2 

779 

2885.2 

829 

3070.4 

879 

3255.6 

929 

3440.7 

979 

3625.9 

780 

2888.9 

830 

3074.1 

880 

3259.3 

930 

3444.4 

980 

3629.6 

781 

2892.6 

831 

3077.8 

881 

3263.0 

931 

3448.2 

981 

3633.3 

782 

2896.3 

832 

3081.5 

882 

3266.7 

932 

3451.9 

982 

3637.0 

783 

2900.0 

833 

3085.2 

883 

3270.4 

933 

3455.6 

983 

3640.7 

784 

2903.7 

834 

3088.9 

884 

3274.1 

934 

3459.3 

984 

3644.4 

785 

2907.4 

835 

3092.6 

885 

3277.8 

935 

3463.0 

985 

3648.2 

786 

2911.1 

836 

3096.3 

886 

3281.5 

936 

3466.7 

986 

3651.9 

787 

2914.8 

837 

3100.0 

887 

3285.2 

937 

3470.4 

987 

3655.6 

788 

2918.5 

838 

3103.7 

888 

3288.9 

938 

3474.1 

988 

3659.3 

789 

2922.2 

839 

3107.4 

889 

3292.6 

939 

3477.8 

989 

3663.0 

790 

2925.9 

840 

3111.1 

890 

3296.3 

940 

3481.5 

990 

3666.7 

791 

2929.6 

841 

3114.8 

891 

3300.0 

941 

3485.2 

991 

3670.4 

792 

2933.3 

842 

3118.5 

892 

3303.7 

942 

3488.9 

992 

3674.1 

793 

2937.0 

843 

3122.2 

893 

3307.4 

943 

3492.6 

993 

3677.8 

794 

2940.7 

844 

3125.9 

894 

3311.1 

944 

3496.3 

994 

3681.5 

795 

2944.4 

845 

3129.6 

895 

3314.8 

945 

snoo.o 

995 

3685.2 

796 

2918.2 

846 

3133.3 

896 

3318.5 

946 

3503.7 

996 

3688.9 

797 

2951.9 

847 

3137.0 

897 

3322.2 

947 

3507.4 

997 

3692.6 

798 

2955.6 

848 

3140.7 

898 

3325.9 

948 

3511.1 

998 

3696.3 

799 

2959.3 

849 

3144.4 

899 

3329.6 

949 

3514.8 

999 

3700.0 

800 

2963.0 

850 

3148.2 

900 

3333.3 

950 

3518.5 

1000 

3703.7 

279 


TABLE  XVI. 


CONVERSION  OF  ENGLISH  INCHES  INTO  CENTIMETRES. 

Ins. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

Cm. 

0 

0.000 

2.540 

5.080 

7.620 

10.16 

12.70 

15.24 

17.78 

20.32 

22.86 

10 

25.40 

27.94 

30.48 

33.02 

35.56 

38.10 

40.64 

43.18 

45.72 

48.26 

20 

50.80 

53.34 

55.88 

58.42 

60.96 

63.50 

66.04 

68.58 

71.12 

73.66 

30 

76.20 

78.74 

81.28 

83.82 

86.36 

88.90 

91.44 

93.98 

96.52 

99.06 

40 

101.60 

104.14 

106.08 

109.22 

111.76 

114.30 

116.84 

119.38 

121.92 

124.46 

50 

127.00 

129.  54  j  132.08 

134.62 

137.16 

139.10 

142.24 

144.78 

147.32 

149.86 

60 

152.40 

154.94    157.48 

160.02 

162.56 

165.10 

167.64 

170.18 

172.72 

175.26 

70 

177.80 

180.34    182.88 

185.42 

187.96 

190.50 

193.04 

195.58 

198.12 

200.96 

80 

203.20 

205.74 

208.28 

210.82 

213.36 

215.90 

218.44 

220.98 

223.52 

226.06 

90 

228.60 

231.14 

233.68 

236.22 

238.76 

241.30 

243.84 

246.38 

248.92 

251.46 

100 

254.00 

256.54 

259.08 

261.62 

264.16 

266.70 

269  24 

271.78 

274.32 

276.8,. 

CONVERSION  OF  CENTIMETRES  INTO  ENGLISH  INCHES. 

Cm. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

0 

0.000 

0.394 

0.787 

1.181 

1.575 

1.969    2.362    2.756 

3.150    3.543 

10 

3.937 

4.331 

4.742     5.118 

5.512 

5.906i  6.299    6.693 

7.087    7.480 

20 

7.874 

8.268 

8.662     9.055 

9.449 

9.843110.236  10.630 

11.024  11.418 

30 

11.811 

12.205 

12.599    12.992 

13.386 

13.780  14.173  14.567  14.961  15.355 

40 
50 

15.748 
19.685 

16.142 
20.079 

16.530    16.929 
20.473    20.867 

17.323 
21.260 

17.717:18.111  18.504  18.898  19.292 
21.  654122.048  22.441122.835  23.229 

60 

23.622 

24.016 

24.410    24.804 

25.197 

25.591 

25  .  985  26  .  378  1  26  772  27  .  1  66 

70 

27.560 

27.953 

28.347    28.741 

29.134 

29.528 

29  .  922  80.  316  80.  709  31  .  103 

80 

31.497 

31.890!  32.284    32.678 

33.071 

33  .  465  33  .  8r>9  34  .  253  <  34  .  646  35  .  040 

90 

35.434 

35.  827  i  36.221    36.615 

37.009 

37  .  402  '  37  .  796  38  .  1  90  38  .  583  38  .  977 

100 

39.370|  39.764    40.158    40.552 

40.945 

41  .339J41  .733  42.126  42.520  42.914 

CONVERSION  OF  ENGLISH  FEET  INTO  METRES. 

Feet. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met. 

Met 

Met. 

0 

0.000 

0.3048 

0.6096 

0.9144 

1.2192 

1.52391.82872.1335 

2.4383 

2.7431 

10 

3.0479 

3.3527 

3.6575 

3.9623 

4.2671 

4.  5719J4.  8767:5.1815 

5.4S63 

5.7911 

20 

6.0359 

6.4006 

6.7055 

7.0102 

7.3150 

7.61987.92468.2294 

8.5342 

8.8390 

30 

9.1438 

9.4486 

9.7534 

10.058 

10.363 

10.668 

10.97211.277 

11.582 

11.887 

40 

12.192 

12.496 

12.801 

13.106 

13.411 

13.716 

14.020!14.325 

14.630 

14.935 

50 

15.239 

15.544 

15.849 

16.154 

16.459 

16.763 

17.068  17.373 

17.678 

17.983 

60 

18.287 

18.592 

18.897 

19.202 

19.507 

19.811 

20.11620.421 

20.726 

21.031 

70 

21.335 

21.640 

21.945 

22.250 

22.555 

22.  859  123.1  64  23.  469 

23.774 

24.079 

80 

24.383 

24.688 

24.993 

25.298 

25.602 

25.907 

26.  212  26.  517126.  822 

27.126 

90 

27.431 

27.736 

28.041 

28.346 

28.651 

28.955 

29.26029.56529.870 

30.174 

100 

30.479 

30.784 

31.089 

31.394 

31.698 

32.003  32.308  32.613  32.918 

33.222 

CONVERSION  OF  METRES  INTO  ENGLISH  FEET. 

Met. 

0 

1 

2 

3 

4 

5 

6 

7 

B' 

9 

Feet. 

Feet. 

Feet,  i  Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

0 

0.000 

3.2809 

6.5618    9.8427 

13.123 

16.404 

19.685 

22.966 

26.247 

29.528 

10 

32.809 

36.090 

39.371    42.651 

45.932 

49.  213i52.494 

55.775 

59.056 

62.337 

20 

65.618 

68.899 

72.179    75.461 

78.741 

82.02285.303 

88.584 

91.865 

95.146 

30 

98.427 

101.71 

104.99    108.27 

111.55 

114.83  118.11 

121.39 

124.67 

127.96 

40 

131.24 

134.52 

137.80   141.08 

144.36 

147.64 

150.92 

154.20 

157.48 

160.76 

50 

164.04 

167.33 

170  61    173.89 

177.17 

180.45 

183.73 

187.01 

190.29 

193.57 

60 

196.85 

200.13 

203.42    206.70 

209.98 

213.26(216.54 

219.82 

223.10 

226.38 

70 

229.66 

232.94 

236.22    239.51 

242.79 

246.07249.35 

252.63 

255.91 

259.19 

8C 

262.47 

265.75 

269.03    272.31 

275.60 

278.88282.16 

^85.44 

288  72 

293.00 

90 

295.28 

298.56 

391.84    305.12 

308.40 

311.69314.97 

318.25 

«1.53 

324.81 

100 

328.09 

331.37 

334.65,  337.93 

341.21 

344.49J  347.  78 

351.06354.34 

357.62 

280 


TABLE  XVH. 


CONVERSION  OF 

ENGLISH  STATUTE-MILES  INTO  KILOMETRES. 

Miles. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

Kilo. 

0 

0.0000 

1.6093 

3.2186 

4.82796.437* 

8.0465 

9.6558 

11.2652 

12.8745 

14.4848 

10 

16.093 

17.70219.312 

20.921  22.531 

24.139 

25.749 

27.358 

28.967 

30.577 

20 

32.186 

33.795 

35.405  37.014 

38.62: 

40.232 

41.842 

43.451 

45.060 

46.670 

30 

48.279 

49.888 

51.49853.107 

54.7K 

56.325 

57.935 

59.544 

61.153 

62.763 

40 

64.372 

65.981  67.591 

69.200 

70.80$ 

1    72.418 

74.028 

75.637 

77.246 

78.856 

50 

80.465 

82.074 

83.684 

85.293 

86.90; 

5    88.511 

90.121 

91  730 

93.339 

94.949 

60 

96.558:98.16799.777 

101.39 

102.  9< 

)    104.60 

106.21 

107.82 

109.43 

111.04 

70 

112.65 

114.26 

115.87 

117.48 

119.  0* 

5    120.69 

122.30 

123.91 

125.52 

127.13 

80 

128.74130.35131.96 

133.57 

135.  1' 

r    136.78 

138.39 

140.00 

141.61 

143.22 

90 

144.85 

146.44 

148.05 

149.66 

151.  2( 

5    152.87 

154.48 

156.09 

157.70 

159.31 

100 

100.93162.53164  14 

165  75 

167.  & 

>    168.96 

170.57 

172.18 

173.79 

175.40 

CONVERSION  OF 

KILOMETRES  INTO  ENGLISH  STATUTE-MILES. 

Kilom. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Miles. 

Miles. 

Miles. 

Miles. 

Miles 

.  Miles. 

Miles. 

Miles. 

Miles. 

Miles. 

0 

0.0000 

0.6214 

1.2427 

1.8641 

2.485 

5    3.1069 

3.7282 

4.3497 

4.9711 

5.5924 

10 

6.2138 

6.8352 

7.4565 

8.0780 

8.699< 

I    9.3208 

9.9421 

10.562 

11.185 

11.805 

20 

12.427 

13.049 

13.670 

14.292 

14.91, 

5    15.534 

16.156 

16.776 

17.399 

18.019 

30 

18.641 

19.263 

19.884 

20.506 

21.12 

r    21.748 

23.370 

22.990 

23.613 

24.233 

40 

24.855 

25.477 

26.098 

26.720 

27.34 

27.962 

28.584 

29.204 

29.827 

30.447 

50 

31.069 

31.690 

32.311 

32.933 

33.55 

I    34.175 

34.797 

35.417 

36.040 

36.660 

60 

37.282 

37.904 

38.525 

39.147 

39.76* 

J    40.389 

41.011 

41.631 

42.254 

42.874 

70 

43.497 

44.118 

44.789 

45.361 

45.98- 

I    46.603 

47.225 

47.845 

48.468 

49.088 

80 

49.711 

50.332 

50.953 

51.575 

52.19 

5    52.817 

53.439 

54.059 

54.682 

55.302 

90 

55.924 

56.545 

57.166 

57.788 

58.40 

)    59.030 

59.652 

60.272 

60.895 

61.515 

100 

62.138 

62.75963.380 

64.002 

64.62 

i    65.244 

65.866 

66.486 

67.109 

67.729 

TABLE   XVIII. 

LENGTH  IN  FEET  OF  1'  ARCS  OF  LATITUDE  AND  LONGITUDE. 

Lat. 

1'  Lat. 

V  Long. 

Lat. 

1'  Lat. 

V  Long. 

1° 

6045 

6085 

31° 

6061 

5222 

2° 

6045 

6083 

32° 

6062 

5166 

3° 

6045 

6078 

33°      N 

6063 

5109 

40 

6045 

6071 

34° 

6064 

5051 

5° 

6045 

6063 

35° 

6065 

4991 

6° 

6045 

6053 

36° 

6066 

4930 

7° 

6046 

6041 

37° 

6067 

4867 

8° 

6046 

6027 

38° 

6068 

4802 

9° 

6046 

6012 

39° 

6070 

4736 

10° 

6047 

5994 

40° 

6071 

4669 

11° 

6047 

5975 

41° 

6072 

4600 

12° 

6048 

5954 

42° 

6073 

4530 

13° 

6048 

5931 

43° 

6074 

4458 

14° 

6049 

5907 

44° 

6075 

4385 

15° 

6049 

5880 

45° 

6076 

4311 

16° 

i 

6050 

5852 

46° 

6077 

4235 

17° 

6050 

5822 

47° 

6078 

4158 

18° 

6051 

5790 

48° 

6079 

4080 

19° 

6052 

-       5757 

49° 

6080 

4001 

20° 

6052 

5721 

50° 

6081 

3920 

21° 

6053 

5684 

51° 

6082 

3838 

22° 

6C54 

5646 

52° 

6084 

3755 

23° 

6054 

5605 

53° 

6085 

3671 

24° 

6055 

5563 

54° 

6086 

3586 

25° 

8056 

£519 

55° 

6087 

3499 

26° 

6057 

5474 

56° 

6088 

3413 

27° 

6058 

5427 

57° 

6089 

3323 

28° 

6059 

5378 

58° 

6090 

3233 

29° 

6060 

5327 

59° 

6091 

3142 

30° 

6061 

5275          I 

60°                    6092 

3051 

281 


EXAMPLE  ILLUSTRATING  USE  OF  TABLE  XIX. 

Find  the  horizontal  distance  and  the  difference  of  level  when 
n=  16°  30',  ak=  580  feet,  and  the  instrumental  constant  c=  .75. 
In  column  headed  16°  opposite  30'  in  the  series  for  "  Horizontal 


TABLE  XIX. 

SHOWING  HORIZONTAL  DISTANCES   AND   DIFFERENCES 
LEVELS  FOR  STADIA  MEASUREMENTS. 


M. 

0° 

1° 

2° 

3° 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

O'.... 

100.00 

.00 

99.97 

1.74 

99.88 

3.49 

99.73 

5.23 

2  

.06 

1.80 

99.87 

3.55 

99.72 

5.28 

4  

" 

.1? 

M 

1.86 

u 

3.60 

99.71 

5.34 

6  

•« 

.17 

99..96 

1.92 

u 

3.66 

" 

5.40 

P  

" 

.23 

1.98 

99.86 

3.72 

99.70 

5.46 

10  

" 

.29 

*' 

2.04 

" 

3.78 

99.69 

5.52 

12  

it 

.35 

M 

2.09 

99.85 

3.84 

u 

5.57 

14  

M 

.41 

99.95 

2.15 

M 

3.90 

99.68 

5.63 

16  

" 

.47 

2.21 

99.84 

3.95 

u 

5.69 

18  

M 

.52 

*' 

2.27 

» 

4.01 

99.67 

5.75 

20  

II 

.58 

" 

2.33 

99.83 

4.07 

99.66 

5.80 

22... 

M 

.64 

99  94 

2.38 

H 

4.13 

M 

5.86 

24  

<« 

.70 

H 

2.44 

99.82 

4.18 

99.65 

5.92 

26  

99.^99 

.76 

II 

2.50 

M 

4.24 

99.64 

5.98 

28  

.81 

99.93 

2.56 

99.81 

4.30 

99.63 

6.04 

80  

" 

.87 

u 

2.62 

4.36 

H 

6.09 

82  

« 

.93 

U 

2.67 

99.80 

4.42 

99.62 

6.15 

84  

M 

.99 

U 

2.73 

4.48 

u 

6.21 

36...  . 

„» 

.05 

99.92 

2.79 

99.J9 

4.53 

99.61 

6.27 

38  

M 

.11 

M 

2.85 

4.59 

99.60 

6.33 

40  

" 

.16 

" 

2.91 

99.78 

4.65 

99.59 

6.38 

42  

M 

.22 

99.91 

2.97 

H 

4.71 

u 

6.44 

44  

99.98 

.28 

M 

3.02 

99.77 

4.76 

99.58 

6.50 

46  

.34 

99.90 

3.08 

4.82 

99.57 

6.56 

48  

« 

.40 

3.14 

99.76 

4.88 

99.56 

6.61 

60...., 

« 

.45 

H 

3.20 

4.94 

" 

6.67 

62  

H 

.51 

99.89 

3.26 

99.75 

4.99 

99.55 

6.73 

54  

« 

.57 

M 

3.31 

99.74 

5.05 

99.54 

6.78 

56  

99.97 

.63 

u 

3.37 

5.11 

99.53 

6.84 

58... 

rt 

.69 

99.88 

3.43 

99.73 

5.17 

99.52 

6.90 

60  

" 

.74 

M 

3.49 

5.23 

99.51 

6.96 

c=  .75 

.75 

.01 

.75 

.02 

.75 

.03 

.75 

.05 

c=1.00 

1.00 

.01 

1.00 

.03 

1.00 

.04 

1.00 

.06 

0=1.25 

1.25 

.02 

1.25 

.03 

1.25 

.05 

1.25 

.08 

From  Winslow's  "Stadia  Surveying."  D.  Van  Nostrantfs  Science  Series. 
282 


Distances,"  we  find  91.93  as  the  expression  for  ak  cos2«  when 
ak=  100  ;  therefore,  when  ak=  580,  alrcosPn  =  91.93  x  5.80  =  533.19. 
Add  to  this  c  cos  n  from  value  of  c  at  bottom  of  page,  and  we  have 
533.19  +  .72  =  533.91,  hor.  dist.  Similarly,  27.23  x  5.80  +  .21  =  157.93, 
diff .  level. 


TABLE  XIX. 

STADIA  MEASUBEMENTS. 


M. 

4° 

5° 

6° 

7* 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

(X  .... 

99.51 

6.96 

99.24 

8.68 

98.91 

1C.  40 

98.51 

12.10 

2  

H 

7.02 

99.23 

8.74 

9G.9C 

10.45 

98.50 

12.15 

4  

99.50 

7.07 

99.22 

8.80 

98.88 

10.U 

98.48 

12.21 

6  

99.49 

7.13 

99.21 

8.85 

98.87 

10.57 

98.47 

12.26 

8  

99.48 

7.10 

99.20 

8.91 

98.86 

10.62 

98.46 

12.32 

10  

99.47 

7.25 

99.19 

8.97 

98.85 

10.68 

98.44 

12.38 

12  

99.46 

7.30 

99.18 

9.03 

98.83 

10.74 

98.43 

12.43 

14  

7.  30 

99.17 

9.08 

98.82 

10-79 

98.41 

12.49 

16  

99.45 

7.42 

99.16 

9.14 

98.81 

10.85 

98.40 

12.55 

18  

yy.44 

7.48 

99.15 

9.20 

98.80 

10.91 

98.39 

12.60 

20  

99.43 

7.53 

99.14 

9.25 

98.78 

10.96 

98.37 

12.66 

22  

99.42 

7.59 

99.13 

9.31 

98.77 

11.02 

98.36 

12.72 

24  

99.41 

7.65 

99.11 

9.37 

98.76 

11.08 

98.34 

12.77 

26  

99.40 

7.71 

99.10 

9.43 

98.74 

11.13 

98.33 

12.83 

28  

99.39 

7.76 

99.09 

9.48 

98.73 

11.19 

98.31 

12.88 

80  

99.38 

7.82 

99.08 

9.54 

98.72 

11.25 

98.29 

12.94 

82... 

99.38 

7.88 

99.07 

9.60 

98.71 

11  .30 

98.28 

13.00 

84  

99.37 

7.94 

99.06 

9.65 

98.69 

11.36 

98.27 

13.05 

86  

99.36 

7.99 

99.05 

9.71 

98.68 

11.42 

98.25 

13.11 

88  

99.35 

8.05 

09.04 

9.77 

98.67 

11.47 

98.24 

13.17 

40  

99.34 

8.11 

99.03 

9.83 

98.65 

11.53 

98.22 

13.22 

42... 

99.33 

8.17 

99.01 

9.88 

98.64 

11.59 

98.20 

13.28 

44.... 

99.32 

8.22 

99.00 

9.94 

98.63 

11.64 

98.19 

13.33 

46  

99.31 

8.28 

98.99 

10.00 

98.61 

11.70 

98.17 

13.39 

48  

99.30 

8.34 

98.98 

10.05 

98.60 

11.76 

98.16 

13.45 

50  

99.29 

8.40 

98.97 

10.11 

98.58 

11.81 

98.14 

13.50 

52  

99.28 

8.45 

98.96 

10.17 

98.57 

11.87 

98.13 

13.58 

54  

99.27 

8.51 

98.94 

10.22 

98.56 

11.93 

98.11 

13.61 

56  

99.26 

8.57 

98.93 

10.28 

98.54 

11.98 

98.10 

13.67 

58  

99.25 

8.63 

98.92 

10.34 

98.53 

12.04 

98.08 

13.73 

60  

99.24 

8.68 

98.91 

10.40 

98.51 

12.10 

98.06 

13.78 

c=  .75 

.75 

.06 

.75 

.07 

.75 

.08 

.74 

.10 

c=1.00 

1.00 

.08 

.99         .09 

.99          .11 

.99 

.13 

c=1.25 

1.25 

.10 

1.24 

.11 

1.24 

.14 

1.24 

.16 

FromWinslow's  "Stadia  Surveying"  1).  Van  ivostrancCs  Science  Series. 

283 


TABLE  XIX. 

STADIA  MEASUREMENTS. 


M. 

8° 

9° 

10° 

11° 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

0'.... 

98.06 

13.78 

97.55 

15.45 

96.98 

17.10 

96.36 

18  73 

2  

98.05 

13.84 

97.53 

15.51 

96.96 

17.16 

96.34 

18.78 

4  

98.03 

13.89 

97.52 

15.56 

96.94 

17.21 

96.32 

IS.  84 

6.... 

98.01 

13.95 

97.50 

15.62 

96.92 

17.26 

96.29 

18  89 

8  

98.00 

14.01 

97.48 

15.  f7 

96.90 

17.32 

96.27 

18.95 

10  

97.98 

14.06 

97.46 

15/3 

96.88 

17.37 

96.25 

19  00 

12  

97.97 

14.12 

97.44 

15  .\  3 

96.86 

17.43 

96.23 

19.05 

14  

97.95 

14.17 

97.43 

15.84 

96.84 

17.48 

96.21 

19.11 

16  

97.93 

14.23 

97.41 

15  89 

96.82 

17.54 

96.18 

19.16 

18  

97.92 

14.28 

97.39 

15.95 

96.80 

17.59 

96.16 

19.21 

20  

97.90 

14.34 

97  37 

16  00 

96.78 

17.65 

96.14 

19  27 

22  

97.88 

14.40 

97.35 

16.06 

96.76 

17.70 

96.12 

19.32 

24  

97.87 

14.45 

97.33 

16.11 

96.74 

17.76 

96.09 

19.38 

26  

97.85 

14.51 

97.31 

16.17 

96.72 

17.81 

96.07 

19.43 

28.... 

97.83 

14.56 

97.29 

16.22 

96.70 

17.86 

96.05 

19.48 

80  

97.82 

14.62 

97.88 

16.  2d 

96.68 

17.92 

96.03 

19.54 

32... 

97.80 

14.67 

97.26 

16.33 

96.66 

17.97 

96.00 

19.59 

84  

97.78 

14.73 

97.24 

16.39 

96.64 

18.03 

95.98 

19.64 

36  

97.76 

14.79 

97.22 

16.44 

96.62 

18.08 

95.96 

19.70 

38  

97.75 

14.84 

97.20 

16.  50 

96.60 

18.14 

95.93 

19.75 

40  

97.73 

14.90 

97.18 

16.55 

96.57 

18.19 

95.91 

19.80 

42  

97.71 

14.  95 

97.16 

16.61 

96.55 

18.24 

95.89 

19.86 

44.  ... 

97.69 

1501 

9T.14 

16.66 

96.53 

18.30 

95.86 

19.91 

46  

97.68 

15.06 

97.12 

16.72 

96.51 

18.35 

95.84 

19.96 

48.... 

97.66 

15.12 

97.10 

16.77 

96.49 

18.41 

95.82 

20.02 

50  

97.64 

15.17 

97.08 

16.83 

96.47 

18.46 

95.79 

20.07 

52  

97.62 

15.23 

97.06 

16.88 

96.45 

18.51 

95.77 

20.12 

54  

97.61 

15.28 

97.04 

16.94 

96.42 

18.57 

95.75 

20.18 

56  .... 

97.59 

15.34 

97.02 

16.99 

96.40 

18.62 

95.72 

20.23 

58... 

97.57 

15.40 

97.00 

17.05 

96.38 

18.68 

95.70 

20.28 

60  

97.55 

15.45 

96.98 

17.10 

96.36 

18.73 

95.68 

20.34 

c=  .75 

.74 

.11 

.74 

.12 

.74 

.14 

.73 

.15 

<r=1.00 

.99         .15 

.99         .16 

.98 

.18 

.98 

.20 

c=1.25 

1.23 

.18 

1.23 

.21 

1.23          .23 

1.22 

.36 

FromWinslow's  "Stadia Surveying."  D.VanNostrantfs Science Seriet. 
284 


TABLE  XIX. 

STADIA  MEASUREMENTS. 


M. 

12° 

13° 

14° 

15° 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Disft. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

0'   .. 

95.68 

20.34 

94.94 

21.92 

94.15 

23.47 

93.30 

25.00 

2  

95.65 

20.39 

94.91 

21.97 

94.12 

23.52 

93.27 

25.05 

4  

95.63 

20.44 

94.89 

22.02 

94.09 

23.58 

93.24 

25.10 

6  

95.61 

20.50 

94  86 

22.08 

94.07 

23.63 

93.21 

25.15 

8  

95.58 

20.55 

94.84 

22.13 

94.04 

23.68 

93.18 

25.20 

10  

95.56 

20.60 

94.81 

22.18 

94.01 

23.73 

93.16 

25.25 

12  

95.53 

20.66 

94.79 

22.23 

93.98 

23.78 

93.13 

25.30 

14  

95.51 

20.71 

94.76 

22.28 

93.95 

23.83 

93.10 

25.35 

16  

95.49 

20.76 

94.73 

22.34 

93.93 

23.88 

93.07 

25.40 

18.  ... 

95.46 

20.81 

94.71 

22.39 

93.90 

23.93 

93.04 

25.45 

20  

95.44 

20.87 

94.68 

22.44 

93.87 

23.99 

93.01 

25.50 

22  

95.41 

20.92 

94.66 

22.49 

93.84 

24.04 

92.98 

25.55 

24  

95.39 

20.97 

94.63 

22.54 

93.81 

24.09 

92.95 

25.60 

26  

95.36 

21.03 

94.60 

22.60 

93.79 

24.14 

92.92 

25.65 

28.  ... 

95.34 

21.08 

94.58 

22.65 

93.76 

24.19 

92.89 

25.70 

30  

95.32 

21.13 

94.55 

22.70 

93.73 

24.24 

92.86 

25.75 

32  

95.29 

21.18 

94.52 

22.75 

93.70 

24.29 

92.83 

25.80 

34  

95.27 

21.24 

94.50 

22.80 

93.67 

24.84 

92.80 

25.85 

86  

95.24 

21.29 

94.47 

22.85 

93.65 

24.39 

92.77 

25.90 

38  

95.22 

21.34 

94.44 

22.91 

93.62 

24.44 

92.74 

25.95 

40  

95.19 

21.39 

94.42 

22.96 

93.59 

24.49 

92.71 

26.00 

42  

95.17 

21.45 

94.39 

23.01 

93.56 

24.55 

92.68 

26.05 

44  

95.14 

21.50 

94.36 

23.06 

93.53 

24.60 

92.65 

26.10 

46  

95.12 

21.55 

94.34 

23.11 

93.50 

24.65 

92.62 

26.15 

48  

95.09 

21.60 

94.31 

23.16 

93.47 

24.70 

92.59 

26.20 

50  

95.07 

21.66 

94.28 

23.22 

93.45 

24.75 

92.56 

26.25 

52... 

95.04 

21.71 

94.26 

23.27 

93.42 

24.80 

92.53 

26.30 

54  

95.02 

21.76 

94.23 

23.32 

93.39 

24.85 

92.49 

26.35 

56  

94.99 

21.81 

94.20 

23.37 

93.36 

24.90 

92.46 

26.40 

58  

94.97 

21.87 

94.17 

23.43 

93.33 

24.95 

92.43 

26.45 

60.... 

94.94 

21.92 

94.15 

23.47 

93.30 

25.00 

92.40 

26.50 

c=  .75 

.73 

.16 

.73          .17 

.73 

.19 

.72          .20 

0=100 

.98          .22 

.97          .23 

.97          .25 

.96          .27 

C=1.25 

1.22 

.27 

1.21 

.20 

1.21 

.81 

1.20 

.34 

"Stadia  Surveying"  D.  ran  Nostr and' s  Science  Series. 
285 


TABLE  XIX. 
STADIA  MEASUKEMENTS. 


M. 

16° 

17° 

18° 

19° 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Sor. 

Diff. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

ist. 

Elev. 

(X.... 

92.40 

26.50 

91.45 

27.96 

90.45 

29.39 

89.40 

30.78 

2  

92.37 

26.55 

91.42 

28.01 

90.42 

29.44 

89.36 

30.83 

4  

92.34 

26.59 

91.39 

28.06 

90.38 

29.48 

89.33 

30.87 

6  

92.31 

26.64 

91.35 

28.10 

90.35 

29.53 

89.29 

30.92 

8  

92.28 

26.69 

91.32 

28.15 

90.31 

29.58 

89.26 

30.97 

10  

92.25 

26.74 

91.29 

28.20 

90.28 

29.62 

89.22 

31.01 

12  

92.22 

26.79 

91.26 

28.25 

90.24 

29.67 

89.18 

31.06 

14  

92.19 

26.84 

91.22 

28.30 

90.21 

29.72 

89.15 

31.10 

16  

92.15 

26.89 

91.19 

28.34 

90.18 

29.76 

89.11 

31.15 

18  

92.12 

26.94 

91.16 

28.39 

90.14 

29.81 

89.08 

31.19 

20  

92.09 

26.99 

91.12 

28.44 

90.11 

29.86 

89.04 

31.24 

28  

92.06 

27.04 

91.09 

28.49 

90.07 

29.90 

89.00 

31.28 

24  

92.03 

27.09 

91.06 

28.54 

90.04 

29.95 

88.96 

31.33 

26  

92.00 

27.13 

91.02 

28.58 

90.00 

30.00 

88.93 

31.38 

28... 

91.97 

27.18 

90.99 

28.63 

89.97 

30.04 

88  89 

31.42 

30  

91.93 

27.23 

90.96 

28.68 

89.93 

30.09 

88.86 

31.47 

32... 

91.90 

27.28 

90.92 

28.73 

89.90 

30.14 

88.82 

31.51 

34  

91.87 

27.33 

90.89 

28.77 

89.86 

30.19 

88.78 

31.56 

36... 

91.84 

27.38 

90  86 

28.82 

89.83 

30.23 

88.75 

31.60 

38  

91.81 

27.43 

90.82 

28.87 

89.79 

30.28 

88.71 

31.65 

40  

91.77 

27.48 

9C-79 

28.92 

89.76 

30.32 

88.67 

31  69 

42  

91.74 

27.52 

90.76 

28.96 

89.72 

30.37 

88.64 

31.74 

44.... 

91.71 

27.57 

90.72 

29.01 

89.69 

30.41 

88.60 

31.78 

46  

91.68 

27.62 

90.69 

29.06 

89.65 

30.46 

88.56 

31.83 

48. 

91.65 

27.67 

90.66 

29.11 

89.61 

30.51 

88.53 

31.87 

50  

91.61 

27.72 

90.62 

29.15 

89.58 

30.55 

88.49 

31.92 

52  

91.58 

27.77 

90.59 

29.20 

89.54 

30.60 

88.45 

31.96 

54  

91.55 

27.81 

90.55 

29.25 

89.51 

30.65 

88.41 

32.01 

56 

91.52 

27.86 

90.52 

29.30 

89.47 

30.69 

88.38 

32.05 

58  

91.48 

27.91 

90.48 

29.34 

89.44 

30.74 

88.34 

32.09 

60  

91.45 

27.96 

90.45 

29.39 

89.40 

30.78 

88.30 

32.14 

c=  .75 

.72 

.21 

.72          .23 

.71          .24 

.71          .25 

<?=1.00 

.96          .28 

.95 

.30 

.95          .32 

.94          .33 

c=1.25 

1.20          .36 

1.19          .38 

1.19 

.40 

1.18 

.42 

From  Winslow's  "Stadia  Surveying."  D.  Van  Xostrand's  Science  Series. 
286 


TABLE  XIX. 

STADIA   MEASUREMENTS. 


M. 

20° 

21° 

22° 

23° 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Dist. 

Elev. 

Dist. 

Elev. 

Dis. 

Elev. 

Dist. 

Elev. 

0'.... 

88.30 

32.14 

87.16 

33.46 

85.97 

34.73 

84.73 

35.97 

2..... 

88.26 

32.18 

87.12 

33.50 

85.93 

34.77 

84.69 

36.01 

4  

88.23 

32.23 

87.08 

33.  64 

85.89 

34.82 

84.65 

36.05 

6  

88.19 

32.27 

87.04 

33.59 

85.85 

34.86 

84.61 

36.09 

8  

88.15 

32.32 

87.00 

33.63 

85.80 

34.90 

84.57 

36.13 

10  

83.11 

32.36 

86.96 

33.67 

85.76 

34.94 

84.52 

36.17 

12  

88.08 

32.41 

86.92 

33.72 

85.72 

34.98 

84.48 

36.21 

14  

88.04 

32.45 

86.88 

33.76 

85.68 

35.02 

84.44 

36.25 

16  

88.00 

32.49 

86.84 

33.80 

85.64 

35.07 

84.40 

36.29 

18  

87.96 

32.54 

86.80 

33.84 

85.60 

35.11 

84.35 

36.33 

20  

87.93 

32.58 

86.77 

33.89 

85.56 

35.15 

84.31 

36.37 

22  

87.89 

32.63 

86.73 

33.93 

85.52 

35.19 

84.27 

36.41 

24  

87.85 

34.67 

86.69 

33.97 

85.48 

35.23 

84.23 

36.45 

26  

87.81 

32.7' 

86.65 

34.01 

85.44 

3527 

84.18 

36.49 

28  

87.77 

32.7<> 

86.61 

34.06 

85.40 

35.31 

84.14 

36.53 

80  

87.74 

32.80 

86.57 

34.10 

85.36 

35.36 

84.10 

36.57 

82  

87.70 

32.85 

86.53 

34.14 

85.31 

35.40 

84.06 

36.61 

34  

87.66 

32.89 

86.49 

34.18 

85.27 

35.44 

84.01 

36.65 

36  

87.62 

32.93 

86.45 

34.23 

85.23 

35.48 

83.97 

36.69 

88  

87.58 

32.93 

86.41 

34.27 

85.19 

35.52 

83.93 

38.73 

40  

87.54 

33.02 

86.37 

34.31 

85.15 

35.56 

83.89 

36.77 

42  

87.51 

33.07 

86.33 

34.35 

85.11 

35.60 

83.84 

36.80 

44  

87.47 

33.11 

86.29 

34.40 

85.07 

35.64 

83.80 

36.84 

46  

87.43 

33.15 

86.25 

34.44 

85.02 

35.68 

83.76 

36.88 

48  

87.39 

33.20 

86.21 

34.48 

84.98 

35.72 

88.72 

36.92 

50  

87.35 

33.24 

86.17 

34.52 

84.94 

35.76 

83.67 

36.96 

52... 

87-31 

33.28 

86.13 

34.57 

84.90 

35.80 

83.63 

37.00 

54  

87.27 

33.33 

86.09 

34.61 

84.86 

35.85 

83.59 

37.04 

56  

87.24 

33.37 

86.05 

34.65 

84.82 

35.89 

83.54 

37.08 

58  

87.20 

33.41 

86.01 

34.69 

84.77 

35.93 

83.50 

37.12 

60  

87.16 

33.46 

85.97 

34.73 

84.73 

35.97 

83.46 

37.16 

e=  .75 

.70          .26 

.70 

.27 

.69          .29 

.69          .30 

c=1.00 

.94 

.35 

.93          .37 

.92          .38 

.92          .40 

c=1.25 

1.17 

.41 

1.16 

.46 

1.15 

.48 

1.15 

.50 

From  Winsloid's  "Stadia  Surveying."  D.  Van  Nostrantfs  Science  Series. 
287 


TABLE  XIX. 

STADIA   MEASUREMENTS. 


M. 

24° 

25° 

26» 

27° 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diflf. 

Hor. 

Diflf. 

Dist. 

Elev. 

Dist. 

Elev 

Dist. 

Elev. 

Dist. 

Elev. 

(X.... 

83.46 

37.16 

82.14 

38.30 

80.78 

39.40 

79.39 

40.45 

2  

83.41 

37.20 

82.09 

38.34 

80.74 

39.44 

79.34 

40.49 

4  

83.37 

37.23 

82.05 

38.38 

80.69 

39.47 

79.30 

40.52 

6  

83.33 

37.27 

82.01 

38.41 

80.65 

39.51 

79.25 

40.55 

8  

83.28 

37.31 

81.96 

38.45 

80.60 

39.54 

79.20 

40.59 

10  

83.24 

37.35 

81.92 

38.49 

80.55 

39.58 

79.15 

40.62 

12..... 

83.20 

37.39 

81.87 

38.53 

80.51 

39.61 

79.11 

40.66 

14  

83.15 

37.43 

81.83 

38.56 

80.46 

39.65 

79.06 

40.69 

16  

83.11 

37.47 

81.78 

38.60 

80.41 

39.69 

79.01 

40.72 

18  

83.07 

37.51 

81.74 

38.64 

80.37 

39.72 

78.96 

40.76 

20  

83.02 

37.54 

81.69 

38.67 

80.32 

39.76 

78.92 

40.79 

22  

82.98 

37.58 

81.65 

38.71 

80.28 

39.79 

78.87 

40.82 

24  

82.93 

37.62 

81.60 

38.75 

80.23 

39.83 

78.82 

40.86 

26  

82.89 

37.66 

81.56 

38.78 

80.18 

39.86 

78.77 

40.89 

28  

82.85 

37.70 

81.51 

38.82 

80.14 

39.90 

78.73 

40.92 

80  

82.80 

37.74 

81.47 

38.86 

80.09 

39.93 

78.68 

40.96 

82  

82.76 

37.77 

81.42 

38.89 

80.04 

39.97 

78.63 

40.99 

82.72 

37.81 

81.38 

38.93 

80.00 

40.00 

78.58 

41.02 

se!'./! 

82.67 

37.85 

81.33 

38.97 

79.95 

40.04 

78.54 

41.06 

88  

8263 

37.89 

81.28 

39.00 

79.90 

40.07 

78.49 

41.09 

40  

82.58 

37.93 

81.24 

39.04 

79.86 

40.11 

78.44 

41.12 

42  

82.54 

37.96 

81.19 

39.08 

79.81 

40.14 

78.39 

41.16 

41  

82.49 

38.00 

81.15 

39.11 

79.76 

40.18 

78.34 

41.19 

46  

82.45 

38.04 

81.10 

39.15 

79.72 

40.21 

78.80 

41.22 

48  

8241 

38.08 

81.06 

39.18 

79.67 

40.24 

78.25 

41.26 

BO  

82.36 

38.11 

81.01 

89.22 

79.62 

40.28 

78.20 

41.29 

S2  

82.32 

38.15 

80.97 

39.26 

79.58 

40.31 

78.15 

41.32 

64  

82.27 

38.19 

80.92 

39.29 

79.53 

40.35 

78.10 

41.35 

56  

82.23 

38.23 

80.87 

39.33 

79.48 

40.38 

78.06 

41.39 

58  

82.18 

38.26 

80.83 

39.36 

79.44 

40.42 

78.01 

41.42 

60  

82.14 

38.30 

80.78 

39.40 

79.39 

40.45 

77.96 

41.45 

C=  .75 

.68          .31 

.68          .32 

.67 

.33 

.66         .85 

C=1.00 

.91 

.41 

.90         .43 

.89          .45 

.89          .48 

C=l  25 

1.14 

.52 

1.13 

.54 

1.12 

.56 

1.11          .58 

FromWinslow's  "Stadia  Surveying."  D.  Van  Nostranffs  Science  Series. 
288 


TABLE  XIX. 

STADIA  MEASUREMENTS. 


M. 

28° 

29° 

30° 

Hor. 

Diff. 

Hor. 

Diff. 

Hor. 

Diff. 

Dist. 

Elev. 

Dist. 

Elev. 

Dist. 

Elev. 

(X.... 

77.96 

41.45 

76.50 

42.40 

75.00 

43.30 

2  

77.91 

41.48 

76.45 

42.43 

74.95 

43.33 

4  

77.86 

41.52 

76.40 

42.46 

74.90 

43  36 

6  

77.81 

41.55 

76.35 

42.49 

74.85 

43.39 

8  

77.77 

41.58 

76.30 

42.53 

74.80 

43.42 

10  

77.72 

41.61 

76.25 

42.56 

74.75 

43.45 

12  

77.67 

41.65 

76.20 

42.59 

74.70 

43.47 

14  

77.62 

41.68 

76.15 

42.62 

74.65 

43.50 

16  

77.57 

41.71 

76.10 

42.05 

74.60 

43.53 

18  

77.52 

41.74 

76.05 

42.68 

74.55 

43.56 

80  

77.48 

41.77 

76.00 

42.71 

74.49 

43.  59 

22  

77.42 

41.81 

75.95 

42.74 

74.44 

43.62 

«4  

77.38 

41.84 

75.90 

42.77 

74.39 

43.65 

26  

77.33 

41.87 

75.85 

42.80 

74.34 

43.67 

28  

77.28 

41.90 

75.80 

42.83 

74.29 

43.70 

BO  

77.23 

41.93 

75.75 

42.86 

74.34 

43.73 

82  

77.18 

41.97 

75.70 

42.89 

74.19 

43  76 

84  

77.13 

42.00 

75.65 

42  92 

74.14 

43.7? 

86  

77.09 

42.03 

75.60 

42.95 

74.09 

43.82 

88  

77.04 

42.06 

75.55 

42.98 

74.04 

43.84 

40  

76.99 

42.09 

75.50 

43.01 

73.99 

43.87 

42  

76.94 

42.12 

75.45 

43.04 

73.93 

43.90 

44  

76.89 

42.15 

T5.40 

43.07 

73.88 

43.93 

46  

76.84 

42.19 

75.35 

43.10 

73.83 

43.95 

48.... 

76.79 

42.22 

75.30 

43.13 

73.78 

43.98 

50  

76.74 

42.25 

75.25 

43.16 

73.73 

44.01 

52  

76.69 

42.28 

75.20 

43.18 

73.68 

44.04 

54.... 

76.64 

42.31 

75.15 

43.21 

73.63 

44.07 

56  

76.59 

42.34 

75.1,0 

43.24 

73.58 

44.09 

58  

76.55 

42.37 

75.05 

43.27 

73.52 

44.12 

60  

76.50 

42.40 

75.00 

43.30 

7347 

44.15 

e=  .75 

.66 

.36 

.65 

.87 

.65         .38 

C=1.00 

.88 

.48 

.87 

.49 

.86         .51 

ff=1.25 

1.10~ 

~^60 

1.09         .62 

1.08 

.64 

From  Winsloufs  "Stadia  Surveying."  D.  Van  Nostrand's  Science  Series. 

289 


TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


1° 


Sine        Cosine 


Sine 


Cosine 


Sine       Cosine 


0 
1 
2 
3 

4 
5 
6 

7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 


6.46373 
76476 
94085 

7.06579 
16270 
24188 


41797 

.46373 
50512 
54291 
57767 


66784 
69417 
71900 
74248 

7.76475 
78594 
80615 
82545 


87870 
89509 


92612 

7.94084 
95508 
96887 
98223 
99520 

8.00779 
02002 
03192 
04350 
05478 

8.06578 
07650 


09718 
10717 
11693 
12647 
13581 
14495 
15391 

8.16968 
17128 
17971 
18798 
19610 
20407 
21189 
21958 
22713 
23456 
24186 


10.00000 
00000 
00000 
00000 
00000 
00000 
00000 
00000 
00000 
00000 

10.00000 
00000 
00000 
00000 
00000 
00000 
00000 
9.99999 


9.99999 
99999 
99999 


99999 
99999 
99999 


99998 


99997 
99997 
99997 


99996 
99996 


99996 
99996 

9.99995 


99995 
99995 
99994 


99994 
99994 
99994 


8.24186   9 


25609 
26304 


27661 
28324 
28977 
29621 


8.30879 
31495 
32103 
32702 


40816 
41307 

8  41792 
42272 
42746 
43216 


44139 
44594 
45044 


45930 

.46366 
46799 
47226 
47650 


48485 
48896 
49304 
49708 
50108 

.50504 
50897 
51287 
51673 
52055 
52434 
52810 
53183 
53552 
53919 
54282 


99993 


99992 
99992 
99992 
99992 


99990 
99990 


33875  99990 
31450  99989 
35018  99989 
35578 
36131 

8.36678 
37217 
37750 
38276 
38796  99987 
99987 


99986 


9998E 


99984 
99983 


99981 


99981 
99980 
999SO 
99979 


99979 
99978 

9.99978 
99977 
99977 
99977 


9997G 
99975 
99975 
99974 
99974 
99974 


8.54282 
54642 
54999 
55354 
55705 
56054 
56400 
56743 
57084 
57421 


58419 
58747 
59072 


59715 


8.60973 
61282 
61589 
61894 
62196 
62497 
62795 


63385 


8.63968 
64256 
64543 
64827 
65110 
65391 
65670 
65947 
66223 
66497 

8.66769 


67308 
67575 
67841 
68104 
68367 
68687 
68886 
69144 

8.69400 
69654 
69907 
70159 
70409 
70658 
70905 
71151 
71395 
71638 
71880 


9  99974 
99973 
99973 
99972 
99972 
69971 
99971 
99970 


99969 


9.99991   8.57757   9.99969 


99967 
99967 


99965 
99964 

9.99964 
99963 
99963 
99962 


99961 


99959 
9.99959 


99958 
99957 
99956 
99956 
99955 
99955 
99954 
99954 

.99953 
99952 
99952 
99951 
99951 
99950 
99949 
99949 
99948 
99948 

.99947 
99946 
99946 
99945 
99944 
99944 


99942 
99942 
99941 


Cosine        Sine 
89° 


Cosine 


Sine 


Cosine 


Sine 


88° 


87C 


TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


40 
41 
42 

43 
44 
45 
46 
47 
48 
49 

50 
51 
5-2 
53 
54 
55 
56 
57 
58 
59 


3° 


Sine   Cosine 


Sine 


Cosine 


Sine 


8  71880 
72120 
72359 
72597 
72834 
73069 
73303 
73535 
73767 
73997 

8  74226 
74454 
74680 
74906 
75130 
75353 
75575 
75795 
76015 
76234 

8.76451 
76667 


77097 
77310 
77522 
77733 
77943 
78152 
78360 

8.78568 
78774 
78979 
79193 
79386 
79588 
79789 
79990 
80189 
80388 

8.80585 
80782 
80978 
81173 
81367 
81560 
81752 
81944 
82134 
82324 

8.82513 
82701 


83075 
83261 
83446 
83630 
83813 
83996 
84177 
84358 


9.99940 
99940 
99939 
99938 


99935 
99934 

.99934 
99933 
99932 
99932 


99928 
99927 


.99926 
99926 


99921 
99920 
99920 

9.99919 
99918 
99917 
99917 


99915 


99913 
99912 


9.99911 


99908 
99907 
99906 
99905 
99904 
99904 


99902 
99901 
99900 


8.84358 
84539 
84718 
84897 
85075 
85252 
85429 


85955 

8.86128 
86301 
86474 
86645 
86816 


87156 
87325 
87494 
87661 

8.87829 
87995 
88161 


88490 


8881' 


89784 
89943 
90102 
90260 
90417 
90574 
90730 
90885 

8.91040 
91195 
91349 
91502 
91655 
91807 


93448 
93594 
93740 


94030 


9.99894 


99887 
99886 

9.99885 


99880 
99879 
99879 


99877 


99875 


99873 
99872 
99871 
99870 


99857 
9.99856 


99854 


99851 


92110  99848 

92261  99847 

92411  99846 

.92561  9.99845 

92710  99844 

92859  99843 

93007  99842 

93154  99841 


99839 


8.94030 
94174 
94317 
94461 


94746 


95728 
95867 
96005 
96143 
96280 
96417 
96553 


8.96825 
96960 
97095 
97229 
97363 
97496 
97629 
97762 

89142     99868     97894 
89304     99867     98026 

8.89464   9.99866   8.98157 
99865 


98419 


98679 

98808 


99194 
99322 

8.99450 
99577 
99704 


99956 

9.00082 

00207 


00456 
00581 

9.00704 
00828 
00951 
01074 
01196 
01318 
01440 
01561 
01682 
01803 
01923 


Cosine       Sine 
86° 


Cosine 


Sine 


Cosine 


Cosine 


99829 

99828 


95029 
95170 
95310 

8.95450   9 


99822 


99817 


99815 
99814 


9.99812 
99810 
99809 


99807 


99802 
99801 

.99800 
99798 
99797 
99796 
99795 
99793 
99792 
99791 
99790 


9.99787 
99786 
99785 
99783 
99782 
99781 
99780 
99778 
99777 
99776 

9.99775 
99773 
99772 
99771 


99767 
99765 
99764 
99763 
99761 


Sine 


85C 
291 


84° 


TABLE  XX.— LOGARITHMIC  SINES  AND  COSINES. 


/ 

6° 

7° 

8" 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.01923 

9.99761 

9.08589 

9.99675 

9.14356 

9.99575 

60 

1 

0*043 

99760 

08692 

99674 

14445 

99574 

59 

2 

02163 

99759 

08795 

99672 

14535 

99572 

58 

3 

02283 

99757 

08897 

99670 

14624 

99570 

57 

4 

02402 

99756 

08999 

99669 

14714 

99568 

56 

5 

02520 

99755 

09101 

99667 

14803 

99566 

55 

6 

02639 

99753 

09202 

99666 

14891 

99565 

54 

7 

02757 

99752 

09304 

99664 

14980 

99563 

53 

8 

02874 

99751 

09405 

99663 

15069 

99561 

52 

9 

02992 

99749 

09506 

99661 

15157 

99559 

51 

10 

9.03109 

9.99748 

9.09606 

9.99659 

9.15245 

9.99557 

50 

11 

03226 

99747 

09707 

99658 

15333 

99556 

49 

12 

03342 

99745 

09807 

99656 

15421 

99554 

48 

13 

03458 

99744 

09907 

99655 

15508 

99552 

47 

14 

03574 

99742 

10006 

99653 

15596 

99550 

46 

15 

03690 

99741 

10106 

99651 

15683 

99548 

45 

16 

03805 

99740 

10205 

99650 

15770 

99546 

44 

17 

03920 

99738 

10304 

99648 

15857 

99545 

43 

18 

04034 

99737 

10402 

99647 

15944 

99543 

42 

19 

04149 

99736 

10501 

99645 

16030 

99541 

41 

20 

9.04262 

9.99734 

9.10599 

9.99643 

9.16116 

9.99539 

40 

21 

04376 

99733 

10697 

99642 

16203 

99537 

39 

22 

04490 

99731 

10795 

99640 

16289 

99535 

38 

23 

04603 

99730 

10893 

99638 

16374 

99533 

37 

24 

04715 

99728 

10990 

99637 

16460 

99532 

36 

25 

04828 

99727 

11087 

99635 

16545 

99530 

35 

26 

04940 

99726 

11184 

99633 

16631 

99528 

34 

27 

05052 

99724 

11281 

99632 

16716 

99526 

33 

28 

05164 

99723 

11377 

99630 

16801 

99524 

32 

29 

05275 

99721 

11474 

99629 

16886 

99522 

31 

30 

9.05386 

9.99720 

9.11570 

9.99627 

9.16970 

9.99520 

30 

31 

05497 

99718 

11666 

99625 

17055 

99518 

29 

32 

05607 

99717 

11761 

99624 

17139 

99517 

28 

33 

05717 

99716 

11857 

99622 

17223 

99515 

27 

34 

05827 

99714 

11952 

99620 

17307 

99513 

26 

a<5 

05937 

99713 

12047 

99618 

17391 

99511 

25 

36 

06046 

99711 

12142 

S9617 

17474 

99509 

24 

37 

06155 

99710 

12236 

99615 

17558 

99507 

23 

38 

06264 

99708 

12331 

99613 

17641 

99505 

22 

39 

06372 

99707 

12425 

99612 

17724 

99503 

21 

40 

9.06481 

9.99705 

9.12519 

9.99610 

9.17807 

9.99501 

20 

41 

06589 

99704 

12612 

99608 

17890 

99499 

19 

42 

06696 

99702 

12706 

99607 

17973 

99497 

18 

43 

06804 

99701 

12799 

99605 

18055 

99495 

17 

44 

06911 

99699 

12892 

99603 

18137 

99494 

16 

45 

07018 

99698 

12985 

99601 

18220 

99492 

15 

46 

07124 

99696 

13078 

99600 

18302 

99490 

14 

47 

07231 

99695 

13171 

99598 

18383 

99488 

13 

48 

07337 

99693 

13263 

99596 

18465 

99486 

12 

49 

07442 

99692 

13355 

99595 

18547 

99484 

11 

50 

9.07548" 

9.99690 

9.13447 

9.99593 

9.18628 

9.99482 

10 

51 

07653 

99689 

13539 

99591 

18709 

99480 

9 

52 

07758 

99687 

13630 

99589 

18790 

99478 

8 

53 

07863 

99686 

13722 

99588 

18871 

99476 

7 

54 

07968 

99684 

13813 

99586 

18952 

99474 

6 

55 

08072 

99683 

13904 

99584 

19033 

99472 

5 

56 

08176 

99681 

13994 

99582 

19113 

99470 

4 

57 

08280 

99680 

14085 

99581 

19193 

99468 

3 

58 

08383 

99678 

14175 

99579 

19273 

99466 

2 

59 

08486 

99677 

14266 

99577 

19353 

99464 

1 

60 

08589 

99675 

14356 

99575 

19433 

99462 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

t 

83° 

82° 

81- 

292 


TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


10° 


Sine       Cosine 


Sine 


Cosine 


Sine       Cosine 


37 


40 
41 
42 
43 
44 
45 
46 
47 
48 
49 

50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


9.19433 
19513 


19672 
19751 


19909 
19988 
20067 
20145 

9.20223 


20380 
20458 
20535 
20613 


20768 
20845 


21076 
21153 
21229 
21306 
21382 
21458 
21534 
21610 
21685 

9.21761 

21836 
21912 
21987 


22137 
22211 


22361 
22435 

9.22509 
22583 
22657 
22731 

22805 
22878 


23025 
23098 
23171 

9.23244 

23317 


28535 
23607 
23679 
23752 


23895 
23967 


9.99462 
99460 
99458 
99456 
99454 
99452 
99450 
99448 
99446 
99444 

9.99442 
99440 


99434 
99432 
99429 
99427 
99425 


9.99421 
99419 
99417 
99415 
99413 
99411 
99409 
99407 
99404 


9.99400 


00302 

99390 
99388 
99385 


9.99379 


99375 
99372 
99370 


99364 


99359 

9.99357 
99355 
99353 
99351 
99:348 
99346 


99337 
99335 


9.23967 
24039 
24110 
24181 
24253 
24324 
24395 
24466 
24536 


9.24677 

24748 


24888 
24958 
25028 
25098 
25168 
25237 
25307. 

9.25376 
25445 
25514 
25583 
25652 
25721 
25790 
25858 


9.26063 
26131 
26199 


26335 
26403 
26470 


26672 

9.26739 
26806 
26873 


27007 
27073 
27140 
27206 
27273 
27339 

.27405 
27471 
27537 
27602 
27668 
27734 
27799 
27864 
27930 
27995 


9.99335 
99333 
99331 
99328 
99326 


99322 
99319 
99317 
99315 


99297 
99294 


99288 
99285 


99281 


99276 
99274 
99271 


9.99267 
99264 


99260 
99257 
99255 
99252 
99250 


99245 


99241 
99238 
99236 
99233 
99231 


99221 


99217 
99214 
99212 
99209 
99207 


99200 
99197 
99195 


9.28060 
28125 
28190 
28254 
28319 


28448 
28512 
28577 
28641 

9.28705 
28769 


29087 
29150 


29277 
.29340 


29529 
129591 
29654 
29716 
29779 
29841 


30028 
30090 
30151 
30213 
30275 


9.30582 
30643 
80704 
30765 


30887 
30947 
31008 
31068 
31129 

9.31189 
31250 
31310 
31370 
31430 
31490 
31549 
31609 


31728 
31788 


9.99195 
99192 
99190 
99187 


99182 


9917? 
99175 
99172 

9.99170 
99167 
99165 
99162 
99160 
99157 
99155 
99152 
99150 
99147 

9.99145 
99142 
99140 
99137 
99135 
99132 
99130 
99127 
99124 
99122 

9.99119 
99117 
99114 
99112 
99109 


99104 
99001 


99078 
99075 
99072 
99070 


99064 
99062 
99059 
99056 
99054 


99043 
99040 


Cosine       Sine 
80° 


Cosine 


Sine 


Cosine 


Sine 


79° 

293 


78° 


TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


f 

12° 

13° 

14° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.31788 

9.99040 

9.35209 

9.98872 

9.38368 

9.98690 

60 

1 

31847 

99038 

35263 

98869 

38418 

98687 

59 

2 

31907 

99035 

35318 

98867 

38469 

98684 

58 

3 

31966 

99032 

35373 

98864 

38519 

98681 

57 

4 

32025 

99030 

35427 

98861 

38570 

98678 

56 

5 

32084 

99027 

35481 

98858 

38620 

98675 

55 

6 

32143 

99024 

35536 

98855 

38670 

98671 

54 

7 

32202 

99022 

35590 

98852 

38721 

986C8 

53 

8 

32261 

99019 

35644 

98849 

38771 

98665 

52 

9 

32319 

99016 

35698 

98846 

38821 

98662 

51 

10 

9.32378 

9.99013 

9.35752 

9.98843 

9.38871 

9.98659 

50 

11 

32437 

99011 

35806 

98840 

38921 

98656 

49 

32 

32495 

99008 

35860 

98837 

38971 

98652 

48 

13 

32553 

99005 

35914 

98834 

39021 

98649 

47 

14 

32612 

99002 

35968 

98831 

39071 

98646 

46 

15 

32670 

99000 

36022 

98828 

39121 

98643 

45 

16 

32728 

98997 

36075 

98825 

39170 

98640 

44 

17 

32786 

98994 

36129 

98822 

39220 

98636 

43 

18 

32844 

98991 

36182 

98819 

39270 

98633 

42 

19 

32902 

98989 

36236 

98816 

39319 

98U30 

41 

20 

9.32960 

9.98986 

9.36289 

9.98813 

9.39369 

9.98627 

40 

21 

33018 

98983 

36342 

98810 

39418 

98623 

39 

22 

33075 

98980 

36395 

98807 

39467 

98620 

38 

23 

33133 

98978 

36449 

98804 

39517 

98617 

37 

24 

33190 

98975 

36502 

98801 

39566 

98614 

36 

25 

33248 

98972 

36555 

98798 

39615 

98610 

86 

26 

33305 

98969 

36608 

98795 

39664 

98607 

34 

27 

33362 

98967 

36660 

98792 

39713 

98604 

33 

28 

33420 

98964 

36713 

98789 

39762 

98601 

32 

29 

33477 

98961 

36766 

98786 

39811 

98597 

31 

30 

9.33534 

9.98958 

9.36819 

9.98783 

9.39860 

9.98594 

30 

31 

33591 

98955 

36871 

98780 

39909 

98591 

29 

32 

33647 

98953 

36924 

98777 

39958 

98588 

28 

33 

33704 

98950 

36976 

98774 

40006 

98584 

27 

34 

33761 

98947 

37028 

98771 

40055 

98581 

26 

35 

33818 

98944 

37081 

98768 

40103 

98578 

25 

36 

33874 

98941 

37133 

98765 

40152 

98574 

24 

37 

33931 

98938 

37185 

98762 

40200 

98571 

23 

38 

33987 

98936 

37237 

98759 

40249 

98568 

22 

39 

34043 

98933 

37289 

98756 

40297 

98565 

21 

40 

9.34100 

9.98930 

9.37341 

9.98753 

9.40346 

9.98561 

20 

41 

34156 

98927 

37393 

98750 

40394 

98558 

19 

42 

34212 

98924 

37445 

98746 

40442 

98555 

18 

43 

34268 

98921 

37497 

98743 

40490 

98551 

17 

44 

34324 

98919 

37549 

98740 

40538 

98548 

16 

45 

34380 

98916 

37600 

98737 

40586 

98545 

15 

46 

34436 

98913 

37652 

98734 

40634 

98541 

14 

47 

34491 

98910 

37703 

98731 

40682 

98538 

13 

48 

34547 

98907 

37755 

98728 

40730 

98535 

12 

49 

34602 

98904 

37806 

98725 

40778 

98531 

11 

50 

9.34658 

9.98901 

9.37858 

9.98722 

9.40825 

9.98528 

10 

51 

34713 

98898 

37909 

98719 

40873 

98525 

9 

52 

34769 

98896 

37960 

98715 

40921 

98521 

8 

53 

34824 

98893 

38011 

98712 

40968 

98518 

f 

54 

34879 

98890 

38062 

98709 

41016 

98515 

6 

55 

34934 

98887 

38113 

98706 

41063 

98511 

5 

56 

34989 

98884 

38164 

98703 

41111 

98508 

4 

57 

35044 

98881 

38215 

98700 

41158 

98505 

3 

58 

35099 

98878 

38266 

98697 

41205 

98501 

2 

59 

35154 

98875 

38317 

98694 

41252 

98498 

1 

60 

35209 

98872 

38368 

98690 

41300 

98494 

0 

] 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

t 

L' 

77° 

76° 

75° 

f 

294 


TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


t 

15° 

16° 

17° 

f 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.41300 

9.98494 

9.44034 

9.98284 

9.46594 

9.98060 

60 

1 

41347 

98491 

44078 

98281 

46635 

98056 

59 

2 

41394 

98488 

44122 

98277 

46676 

98052 

58 

3 

41441 

98484 

44166 

98273 

46717 

98048 

57 

4 

41488 

98481 

44210 

98270 

46758 

98044 

56 

5 

41535 

98477 

44253 

98266 

46800 

98040 

55 

6 

41582 

98474 

44297 

98262 

46841 

98036 

54 

7 

41628 

98471 

44341 

98259 

46882 

98032 

53 

8 

41675 

98467 

44385 

98255 

46923 

98029 

52 

9 

41722 

98464 

44428 

98251 

46964 

98025 

51 

10 

9.41768 

9.98460 

9.44472 

9.98248 

9.47005 

9.98021 

50 

11 

41815 

98457 

44516 

98244 

47045 

98017 

49 

12 

41861 

98453 

44559 

98240 

47086 

98013 

48 

13 

41908 

98450 

44602 

98237 

47127 

98009 

47 

14 

41954 

98447 

44646 

98233 

47168 

98005 

46 

15 

42001 

98443 

44689 

98229 

47209 

98001 

45 

16 

42047 

98440 

44733 

98226 

47249 

97997 

44 

17 

42093 

98436 

44776 

98222 

47290 

97993 

43 

18 

42140 

98433 

44819 

98218 

47330 

97989 

42 

19 

42186 

98429 

44862 

98215 

47371 

97986 

41 

20 

9.4-2232 

9.98426 

9.44905 

9.98211 

9.47411 

9.97982 

40 

21 

42278 

98422 

44948 

98207 

47452 

97978 

39 

22 

42324 

98419 

44992 

98204 

47492 

97974 

38 

23 

42370 

98415 

45035 

98200 

47533 

97970 

37 

24 

42416 

98412 

45077 

98196 

47573 

97966 

36 

25 

42461 

98409 

45120 

98192 

47613 

97962 

35 

26 

42507 

98405 

45163 

98189 

47654 

97958 

34* 

27 

42553 

98402 

45206 

98185 

47694 

97954 

33 

28 

42599 

98398 

45249 

98181 

47734 

97950 

32 

29 

42644 

98395 

45292 

98177 

47774 

97946 

31 

30 

9.42690 

9.98391 

9.45334 

9.98174 

9.47814 

9.97942 

30 

31 

42735 

98388 

45377 

98170 

47854 

97938 

29 

32 

42781 

98384 

45419 

98166 

47894 

97934 

28 

33 

42826 

98381 

45462 

98162 

47934 

97930 

27 

34 

42872 

98377 

45504 

98159 

47974 

97926 

26 

35 

42917 

98373 

45547 

98155 

4K)14 

97922 

25 

36 

42962 

98370 

45589 

98151 

48054 

97918 

24 

37 

43008 

98366 

45632 

98147 

48094 

97914 

23 

38 

43053 

98363 

45674 

98144 

48133 

97910 

22 

39 

43098 

98359 

45716 

98140 

48173 

97906 

21 

40 

9.43143 

9.98356 

'9.45758 

9.98136 

9.48213 

9.97902 

20 

41 

43188 

98352 

45801 

98132 

48252 

97898 

19 

42 

43233 

98349 

45843 

98129 

48292 

97894 

18 

43 

43278 

98345 

45885 

98125 

48332 

97890 

17 

44 

43323 

98342 

45927 

98121 

48371 

97886 

16 

45 

43367 

98338 

45969 

98117 

48411 

97882 

15 

46 

43412 

98334 

46011 

98113 

48450 

97878 

14 

47 

43457 

98331 

46053 

98110 

48490 

97874 

13 

48 

43502 

98327 

46095 

98106 

48529 

97870 

12 

49 

43546 

98324 

46136 

98102 

48568 

97866 

11 

50 

9.43591 

9.98320 

9.46178 

9.98098 

9.48607 

9.97861 

10 

51 

43635 

98317 

46220 

98094 

48647 

97857 

9 

52 

43680 

98313 

46262 

98090 

48686 

97853 

8 

53 

43724 

98309 

46303 

98087 

48725 

97849 

7 

54 

43769 

98306 

46345 

98083 

48764 

97845 

6 

55 

43813 

98302 

46386 

98079 

48803 

97841 

5 

56 

43857 

98299 

46428 

98075 

48842 

97837 

4 

57 

43901 

98295 

46469 

98071 

48881 

97833 

3 

58 

43946 

98291 

46511 

98067 

48920 

97829 

2 

59 

43990 

98288 

46552 

98063 

48959 

97825 

i 

60 

44034 

98284 

46594 

98060 

48998 

97821 

0 

, 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

i 

74° 

73° 

72° 

295 


TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


/ 

18° 

19° 

20° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.48998 

9.97821 

9.51264 

9.97567 

9.53405 

9.97299 

60 

1 

49037 

97817 

51301 

97563 

53440 

97294 

59 

2 

49076 

97812 

51338 

97558 

53475 

97289 

58 

3 

49115 

97808 

51374 

97554 

5:3509 

97285 

57 

4 

49153 

97804 

51411 

97550 

53544 

97280 

56 

5 

49192 

97800 

51447 

97545 

53578 

97276 

55 

6 

49231 

97796 

51484 

97541 

53613 

97271 

54 

7 

49269 

97792 

51520 

97536 

53647 

97266 

53 

8 

49308 

97788 

51557 

97532 

53682 

97262 

52 

9 

49347 

97784 

51593 

97528 

53716 

97257 

51 

10 

9.49385 

9.97779 

9.51629 

9.97523 

9.53751 

9.97252 

50 

11 

49424 

97775 

51666 

97519 

53785 

97248 

49 

12 

49462 

97771 

51702 

97515 

53819 

97243 

48 

13 

49500 

97767 

51738 

97510 

53854 

97238 

47 

14 

49539 

97763 

51774 

97506 

53888 

97234 

46 

15 

49577 

97759 

51811 

97501 

53922 

97229 

45 

16 

49615 

97754 

51847 

97497 

53957 

97224 

44 

17 

49654 

97750 

51883 

97492 

53991 

97220 

43 

18 

49692 

97746 

51919 

97488 

54025 

97215 

42 

19 

49730 

97742 

51955 

97484 

54059 

97210 

41 

20 

9.49768 

9.97738 

9.51991 

9.97479 

9.54093 

9.97206 

40 

21 

49806 

97734 

52027 

97475 

54127 

97201 

39 

22 

49844 

97729 

52063 

97470 

54161 

97196 

38 

23 

49882 

97725 

52099 

97466 

54195 

97192 

37 

24 

49920 

97721 

52135 

97461 

54229 

97187 

36 

25 

49958 

9T717 

52171 

97457 

54263 

97182 

35 

26 

49996 

97713 

52207 

97453 

54297 

97178 

34 

27 

50034 

97708 

52242 

97448 

54331 

97173 

33 

28 

50072 

97704 

52278 

97444 

54365 

97168 

32 

29 

50110 

97700 

52314 

97439 

54399 

97163 

31 

30 

9.50148 

9.97696 

9.52350 

9.97435 

9.54433 

9.97159 

30 

31 

50135 

97691 

52385 

97430 

54466 

97154 

29 

32 

50223 

97687 

52421 

97426 

54500 

97149 

28 

33 

50261 

97683 

52456 

97421 

51534 

97145 

27 

34 

50298 

97679 

52492 

97417 

54567 

97140 

26 

35 

50336 

97674 

52527 

97412 

54601 

97135 

25 

36 

50374 

97670 

5*663 

97408 

54635 

97130 

24 

37 

50411 

97666 

52598 

97403 

54668 

97126 

23 

38 

50449 

97662 

52634 

97399 

54702 

97121 

22 

39 

50486 

97657 

52669 

97394 

54735 

97116 

21 

40 

9.50523 

9.97653 

9.52705 

9.97390 

•  9.54769 

9.97111 

20 

41 

50561 

976*9 

52740 

97385 

54802 

97107 

19 

42 

5059S 

97645 

52775 

97381 

54836 

97102 

18 

43 

50635 

97640 

52811 

97376 

54869 

97097 

17 

44 

50673 

97636 

52846 

97372 

54903 

97092 

16 

45 

50710 

97632 

52881 

97367 

54936 

97087 

15 

46 

50747 

97628 

52916 

97363 

54969 

97083 

14 

47 

50784 

97623 

52951 

97358 

55003 

97078 

13 

48 

50821 

97619 

52986 

97353 

55036 

97073 

12 

49 

50858 

97615 

53021 

97349 

55069 

97068 

11 

50 

9.50896 

9.97610 

9.53056 

9.97344 

9.55102 

9.97063 

10 

51 

50933 

97606 

53092 

97340 

55136 

97059 

9 

52 

50970 

97602 

53126 

97335 

55169 

97054 

8 

53 

51007 

97597 

53161 

97331 

55202 

97049 

7 

54 

51043 

97593 

53196 

97326 

55235 

97044 

6 

55 

51080 

97589 

53231 

97322 

55268 

97039 

5 

56 

51117 

97584 

53266 

97317 

55301 

97035 

4 

57 

51154 

97580 

53301 

97312 

55334 

97030 

3 

58 

51191 

97576 

53336 

97308 

55367 

97025 

2 

59 

51227 

97571 

53370 

97303 

55400 

97020 

1 

60 

51264 

97567 

53405 

97299 

55433 

97015 

0 

i 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

71° 

70° 

69° 

296 


TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


; 

21° 

22° 

23 

' 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

o 

9.55433 

9.97015 

9.57358 

9.96717 

9.59188 

9.96403  \  60 

i 

55466 

97010 

57389 

96711 

59218 

96397 

59 

2 

55499 

97005 

57420 

96706 

59247 

96392 

58 

3 

55532 

97001 

57451 

96701 

59277 

963  "7 

57 

4 

55564 

96996 

57482 

96696 

59307 

96381 

56 

5 

55597 

96991 

57514 

96691 

59336 

96376 

55 

6 

55630 

96986 

57545 

96686 

59366 

96370 

54 

7 

55663 

96981 

57576 

96681 

59396 

9(5365 

53 

8 

55695 

96976 

57607 

96676 

59425 

96360 

52 

9 

55728 

96971 

57638 

96670 

59455 

96354 

51 

10 

9.55761 

9.96966 

9.57669 

9.96665 

9.59484 

9.96349 

50 

11 

55793 

96962 

57700 

96060 

59514 

96343 

49 

12 

55826 

96957 

57731 

96655 

59543 

96338 

48 

13 

55858 

96952 

57762 

96650 

59573 

96333 

47 

14 

55891 

96947 

57793 

96645 

59602 

96327 

46 

15 

55923 

96942 

57824 

96640 

59632 

96322 

45 

16 

55956 

96937 

57855 

96634 

59661 

96316 

44 

17 

55988 

96932 

57885 

96629 

59690 

96311 

43 

18 

56021 

96927 

57916 

96624 

59720 

96305 

42 

19 

56053 

96922 

57947 

96619 

59749 

96300 

41 

20 

9.56085 

9.96917 

9.57978 

9.96614 

9.59778 

9.96294 

40 

21 

56118 

96912 

58008 

96608 

59808 

96289 

39 

22 

56150 

96907 

58039 

96603 

59837 

96284 

38 

23 

56182 

96903 

58070 

96598 

59866 

96278 

37 

24 

56215 

96898 

58101 

96593 

59895 

96273 

36 

25 

56247 

96893 

58131 

96588 

59924 

96267 

35 

26 

56279 

96888 

58162 

96582 

59954 

96262 

34 

27 

56311 

96883 

58192 

96577 

59983 

96256 

33 

28 

56343 

96878 

58223 

96572 

60012 

96251 

32 

29 

56375 

96873 

58253 

96567 

60041 

96245 

31 

30 

9.56408 

9.96868 

9.58284 

9.96562 

9.60070 

9.96240 

30 

31 

56440 

96863 

58314 

96556 

G0099 

96234 

29 

32 

56472 

96858 

58345 

96551 

60128 

96229 

28 

33 

56504 

96853 

58375 

96546 

60157 

96223 

27 

34 

56536 

96848 

58406 

96541 

60186 

Mi  18 

26 

35 

56568 

96843 

58436 

96535 

60215 

96212 

25 

36 

37 

56599 
56631 

9G838 
96833 

58467 
58497 

96530 
96525 

60244 
60273 

96207 
96201 

24 
23 

38 

56663 

96828 

58527 

96520 

60302 

96196 

22 

39 

56695 

96823 

58557 

96514 

60331 

96190 

21 

40 

9.56727 

9.96818 

9.58588 

9.96509 

9.60359 

9.96185 

20 

41 

'  56759 

96813 

58618 

96504 

60388 

96179 

19 

42 

56790 

96808 

58648 

96498 

60417 

96174 

18 

43 

56822 

96803 

58678 

96493 

60446 

96168 

17 

44 

56854 

96798 

58709 

96488 

60474 

96162 

16 

45 

56886 

96793 

58739 

96483 

60503 

96157 

15 

46 

56917 

96788 

58769 

96477 

60532 

96151 

14 

47 

56949 

96783 

58799 

96472 

60561 

96146 

13 

48 

56980 

96778 

58829 

96467 

60589 

96140 

12 

49 

57012 

96772 

58859 

96461 

60618 

96135 

11 

50 

9.57044 

9.96767 

9.58889 

9.96456 

9.60646 

9.96129 

10 

51 

57075 

96762 

58919 

96451 

60675 

96123 

9 

52 

57107 

96757 

58949 

96445 

60704 

96118 

g 

53 

57138 

96752 

58979 

96440 

60732 

96112 

7 

54 

57169 

96747 

59009 

96435 

60761 

96107 

6 

55 

57201 

96742 

59039 

96429 

60789 

96101 

5 

56 

57232 

96737 

59069 

96424 

60818 

96095 

4 

57 

57264 

96732 

59098 

96419 

60846 

96090 

g 

58 

57295 

96727 

59128 

96413 

60875 

960S4 

2 

59 

57326 

96722 

59158 

96408 

60903 

96079 

60 

57358 

96717 

59188 

96403 

60931 

96073 

0 

t 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

§ 

68° 

67° 

66° 

297 


TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


/ 

24° 

25° 

26° 

/ 

Sit.e 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.60931 

9.96073 

9.62595 

9.95728 

9.G4184 

9.95366 

60 

1 

60960 

96067 

62622 

95722 

64210 

95360 

59 

2 

6098S 

960G2 

62649 

95716 

64236 

95354 

58 

3 

61016 

9G056 

62676 

95710 

642G2 

95348 

57 

4 

61045 

96050 

62703 

95704 

64288 

95341 

56 

5 

61073 

96045 

62730 

95G98 

64313 

95335 

55 

6 

61101 

96039 

62757 

95692 

64339 

95329 

54 

7 

61129 

96034 

62784 

95686 

64365 

95323 

53 

8 

61158 

9(3028 

62811 

956SO 

64391 

95317 

52 

9 

61186 

96022 

62838 

95674 

64417 

95310 

51 

10 

9.61214 

9.96017 

9.62865 

9.95668 

9.64442 

9.95304 

50 

11 

61242 

96011 

62892 

95663 

64468 

95298  i  49 

12 

61270 

96005 

62918 

95657 

64494 

95292 

48 

13 

61*98 

96000 

62945 

95651 

64519 

95286 

47 

14 

61326 

95994 

62972 

95645 

64545 

95279 

46 

15 

61354 

95988 

62999 

95639 

64571 

95273 

45 

16 

61382 

95982 

63026 

95633 

64596 

95267 

44 

17 

61411 

95977 

63052 

95627 

64622 

95261 

43 

18 

61438 

95971 

63079 

95621 

64647 

95254 

42 

19 

6  1466 

95965 

63106 

95615 

64673 

95248 

41 

20 

9.61494 

9.95960 

9.63133 

9.95609 

9.64698 

9.95242 

40 

21 

61522 

95954 

63159 

95603 

64724 

952^6 

39 

22 

61550 

95948 

63186 

95597 

64749 

95229 

38 

23 

61578 

95942 

63213 

95591 

64775 

95223 

37 

24 

61606 

95937 

63239 

95585 

64800 

95217 

36 

25 

61634 

95931 

63266 

95579 

64826 

95211 

35 

26 

61662 

95925 

63292 

95573 

64851 

95204 

34 

27 

61689 

95920 

63319 

95567 

C4877 

95198 

33 

28 

61717 

95914 

63345 

95561 

64902 

95192 

32 

29 

61745 

95908 

63372 

95555 

64927 

95185 

31 

30 

9.61773 

9.95902 

9.63398 

9.95549 

9.64953 

9.95179 

30 

31 

61800 

95897 

63425 

95543 

64978 

95173 

29 

32 

61828 

95891 

63451 

95537 

65003 

95167    28 

33 

61856 

95885 

63478 

95531 

65029 

95160    27 

34 

61883 

95879 

63504 

95525 

65054 

95154  !  26 

35 

61911 

95873 

63531 

95519 

65079 

95148    25 

36 

61939 

95S68 

63557 

95513 

65104 

95141    24 

37 

61966 

95862 

635&3 

95507 

65130 

95135    23 

38 

61994 

95856 

63610 

95500 

65155 

95129 

22 

39 

62021 

95850 

63636 

95494 

65180 

95122 

21 

40 

9.6-2049 

9.95844 

9.63662 

9.95488 

9.65205 

9.95116 

20 

41 

62076 

95839 

63689 

95482 

65230 

95110 

19 

42 

62104 

95833 

63715 

95476 

65255 

95103 

18 

43 

62131 

95827 

63741 

95470 

65281 

95097 

17 

44 

62159 

95821 

63767 

95464 

65306 

95090 

16 

45 

62186 

95815 

63794 

95458 

65331 

95084 

15 

46 

62214 

95810 

63820 

95452 

65356 

1)5078 

14 

47 

62241 

95804 

63846 

95446 

65381 

95071 

13 

48 

62268 

95798 

63S72 

95440 

65406 

95065 

12 

49 

62296 

95792 

63898 

95434 

65431 

95059 

11 

50 

9.62323 

9.95786 

9.63924 

9.95427 

9.65456 

9.95052 

10 

51 

62350 

95780 

63950 

95421 

65481 

95046 

9 

52 

62377 

95775 

63976 

95415 

65506 

95039 

8 

53 

62405 

95769 

64002 

95409 

65531 

95033 

7 

54 

62432 

95763 

64028 

95403 

65565 

95027 

6 

55 

62459 

95757 

64054 

95397 

65580 

95020 

5 

56 

62486 

95751 

64080 

95391 

65605 

95014 

4 

57 

62513 

95745 

64106 

95384 

65630 

95007 

3 

58 

62541 

95739 

64132 

95378 

65655 

95001 

2 

59 

62568 

95733 

64158 

95372 

65680 

94995 

1 

60 

62595 

95728 

64184 

95366 

65705 

949«8 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

66° 

64° 

63° 

298 


TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


27° 

28° 

29° 

f 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

1 

9.65705 

9.94988 

9.G7161 

9.94593 

9.68557 

9.94182 

60 

1 

65729 

94982 

67186 

94587 

68580 

94175 

59 

2 

65754 

94975 

67208 

94580 

68603 

94168 

58 

3 

65779 

94969 

67232 

94573 

68625 

94161 

57 

4 

65804 

94962 

67256 

94567 

68648 

94154 

56 

5 

65828 

94956 

67280 

94560 

68G71 

94147 

55 

6 

65853 

94949 

67303 

94553 

68694 

94140 

54 

7 

65878 

94943 

67327 

94546 

68716 

94133 

53 

8 

65902 

94936 

67350 

94540 

68739 

94126 

52 

9 

G5927 

94930 

67374 

94533 

68762 

94119 

51 

10 

9.65952 

9.94923 

9.67398 

9.94526 

9.68784 

9.94112 

50 

11 

65976 

94917 

67421 

94519 

68807 

94105 

W 

12 

66001 

94911 

67445 

94513 

68829 

94098 

48 

13 

66025 

94904 

67468 

94506 

68852 

94090 

47 

14 

66050 

94898 

67492 

94499 

68875 

94083 

46 

15 

66075 

94891 

67515 

94492 

68897 

94076 

45 

16 

66099 

94885 

67539 

94485 

68920 

940G9 

44 

17 

66124 

94878 

67562 

94479 

68942 

94062 

43 

18 

66148 

94871 

67586 

94472 

68965 

94055 

42 

19 

66173 

948G5 

67609 

94465 

68987 

94048 

41 

20 

9.66197 

9.94858 

9.67633 

9.94458 

9.69010 

9.94041 

40 

21 

66221 

94852 

67656 

94451 

69032 

94034 

39 

22 

66246 

94S45 

67680 

94445 

69055 

94027 

38 

23 

66-J70 

94839 

67703 

94438 

69077 

94020 

37 

24 

66295 

94832 

67726 

94431 

69100 

94012 

36 

25 

66319 

94826 

67750 

94424 

69122 

94005 

35 

26 

66343 

94819 

67773 

94417 

69144 

93998 

34 

27 

66368 

94813 

67796 

94410 

69167 

93991 

33 

28 

66302 

94806 

67820 

94404 

69189 

93984 

32 

29 

66416 

94799 

67843 

94397 

69212 

93977 

31 

30 

9.66441 

9.94793 

9.67866 

9.94390 

9.69234 

9.93970 

30 

31 

66465 

94786 

67890 

94383 

69256 

93963 

29 

32 

66489 

94780 

67913 

94376 

69279 

93955 

28 

33 

66513 

94773 

67936 

94369 

69301 

93948 

27 

34 

66537 

947G7 

67959 

94362 

69323 

93941 

26 

35 

665G2 

94760 

67982 

94355 

69345 

93934 

25 

36 

66K6 

94753 

68006 

94349 

69368 

93927 

24 

37 

66610 

94747 

C8029 

94342 

69390 

93920 

23 

38 

66634 

947'40 

68052 

94335 

69412 

93912 

22 

39 

66G58 

94734 

68075 

94328 

69434 

93905 

21 

40 

9.666^2 

9.94727 

9.68098 

9.94321 

9.69456 

9.93898 

20 

41 

66706 

94720 

68121 

94314 

69479 

93891 

19 

42 

66731 

94714 

68144 

94307 

69501 

93884 

18 

43 

66755 

94707 

68167 

94300 

69523 

93876 

17 

44 

66779 

94700 

68190 

94293 

69545 

93869 

16 

45 

66803 

94694 

68213 

94286 

69567 

93862 

15 

46 

66827 

94687 

68237 

94279 

69589 

938f,5 

14 

47 

66851 

94680 

68260 

94273 

69611 

93847 

13 

48 

66875 

94674 

68283 

94266 

696:^3 

93840 

12 

49 

66899 

94G67 

68305 

94259 

69655 

93833 

11 

50 

9.66922 

9.94660 

9.68328 

9.94252 

9.69677 

9.93826 

JO 

51 

66946 

94654 

68351 

94245 

69G99 

93819 

9 

52 

66970 

94647 

68374 

94238 

69721 

93811 

8 

53 

66994 

94640 

68397 

94231 

69743 

93804 

7 

54 

67018 

94634 

68420 

94224 

69765 

93797 

6 

55 

67042 

94627 

68443 

94217 

69787 

93789 

5 

56 

67066 

94620 

68466 

94210 

69809 

93781 

4 

57 

67090 

94614 

68489 

94203 

69831 

93775 

3 

58 

67113 

94607 

68512 

94196 

69853 

93768 

2 

59 

67137 

94600 

68534 

94189 

69875 

93760 

1 

60 

67161 

94593 

68557 

94182 

69897 

93753 

0 

f 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

\ 

< 

82° 

61° 

60° 

TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


30° 

31 

» 

82° 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.69897 

9.93753 

9.71184 

9.93307 

9.72421 

9.92842 

60 

1 

69919 

93746 

71205 

93299 

72441 

92834 

59 

2 

69941 

93738 

71226 

93291 

72461 

92826 

58 

3 

69963 

93731 

71247 

93284 

72482 

92818 

57 

4 

69984 

93724 

71268 

93276 

72502 

92810 

56 

5 

70006 

93717 

71289 

93269 

72522 

92803 

55 

6 

70028 

93709 

71310 

93261 

72542 

92795 

54 

7 

70050 

93702 

71331 

93253 

72562 

92787 

53 

8 

70072 

93095 

71352 

93246 

72582 

92779 

52 

9 

70093 

93687 

71373 

93238 

72602 

92771 

51 

10 

9.70115 

9.»]680 

9.71393 

9.93230 

9.72622 

9.92763 

50 

11 

70137 

93673 

71414 

93&.'3 

72643 

92755 

49 

12 

70159 

93665 

71435 

93215 

72663 

92747 

48 

13 

70180 

93658 

71456 

93207 

72683 

92739 

47 

14 

70202 

93650 

71477 

93-200 

72703 

92731 

46 

15 

70224 

93643 

71498 

93192 

72723 

92723 

45 

16 

70245 

93636 

71519 

93184 

72743 

92715 

44 

17 

70267 

93628 

71539 

93177 

72763 

92707 

43 

18 

70288 

93621 

71560 

93169 

72783 

9-2699 

42 

19 

70310 

93614 

71581 

93161 

72803 

92691 

41 

20 

9.70332 

9.93606 

9.71602 

9.93154 

9.72823 

9.92683 

40 

21 

70353 

93599 

71622 

93146 

72843 

9267'5 

39 

22 

70375 

93591 

71643 

93138 

72863 

92667 

38 

23 

70396 

93584 

71664 

93131 

72883 

92659 

37 

24 

70418 

93577 

71685 

93123 

72902 

92651 

36 

25 

70439 

93569 

71705 

93115 

72922 

92643 

35 

26 

70461 

93562 

71726 

93108 

72942 

92635 

34 

27 

70482 

93554 

71747 

93100 

72962 

92627 

33 

28 

70504 

93547 

71767 

93092 

72982 

92619 

32 

29 

70525 

93539 

71788 

93084 

73002 

92611 

31 

30 

9.70547 

9.93532 

9.71809 

9.93077 

9.73022 

9.92603 

30 

31 

70568 

93525 

71829 

93069 

73041 

92595 

29 

38 

70590 

93517 

71850 

93061 

73061 

92587 

28 

33 

70611 

93510 

71870 

93053 

73081 

92579 

27 

34 

70633 

93502 

71891 

93046 

73101 

92571 

26 

35 

70654 

93495 

71911 

93038 

73121 

92563 

25 

36 

70675 

93487 

71932 

93030 

73140 

92555 

24 

87 

70697 

93480 

71952 

93022 

73160 

92546 

23 

38 

70718 

93472 

71973 

93014 

73180 

9-2538 

22 

39 

70739 

93465 

71994 

93007 

73200 

92D30 

21 

40 

9.70761 

9.93457 

9.72014 

9.92999 

9.73219 

9.92522 

20 

41 

70782 

93450 

72034 

92991 

73239 

92514 

19 

42 

70803 

93442 

72055 

92983 

73259 

92506 

18 

43 

70824 

93435 

72075 

92976 

73278 

9-2498 

17 

44 

70846 

93427 

72096 

92968 

73298 

92490 

16 

45 

70867 

93420 

72116 

92960 

73318 

92482 

15 

46 

70888 

93412 

72137 

92952 

73337 

92473 

14 

47 

70909 

93405 

72157 

92944 

78857 

92465 

13 

48 

70931 

93397 

72177 

92936 

73377 

92457  I  12 

49 

70952 

93390 

72198 

92929 

73396 

92449    11 

50 

9.70973 

9.93382 

9.72218 

9.92921 

9.73416 

9.92441 

10 

51 

70994 

93375 

72238 

92913 

73435 

92433 

9 

53 

71015 

93367 

72259 

9-2905 

73455 

92425 

8 

53 

71036 

93360 

72279 

92897 

73474 

92416 

7 

54 

71058 

93352 

72299 

92889 

73494 

92408 

6 

55 

71079 

93344 

72320 

92881 

73513 

92400 

5 

56 

71100 

93337 

72340 

92874 

73533 

92392 

4 

57 

71121 

93329 

72360 

92866 

73552 

92384 

3 

58 

71142 

93322 

72381 

92858 

73572 

92376 

2 

59 

71163 

93314 

72401 

92850 

73591 

92367 

1 

60 

71184 

93307 

72421 

92842 

73611 

9-2359 

0 

/ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

/ 

69° 

J 

8° 

57° 

TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


33° 

84° 

35 

0 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.73611 

9.92359 

9.74756 

9.91857 

9.75859 

9.91336 

60 

1 

73630 

92351 

74775 

91849 

75877 

91328 

59 

2 

73650 

92343 

74794 

91840 

75895 

91319 

58 

3 

73669 

92335 

74812 

91832 

75913 

91310 

57 

4 

73689 

92326 

74831 

91823 

75931 

91301 

56 

5 

73708 

92318 

74850 

91815 

75949 

91292 

55 

6 

73727 

92310 

74868 

91806 

75967 

91283 

54 

7 

73747 

92302 

74887 

91798 

75985 

91274 

53 

8 

73766 

92293 

74906 

91789 

76003 

91266 

52 

9 

73785 

92285 

74924 

91781 

76021 

91257 

51 

10 

9.73805 

9.92277 

9.74943 

9.91772 

9.76039 

9.91248 

50 

11 

73824 

92269 

74961 

91763 

76057 

91239 

49 

12 

73843 

92260 

74980 

91755 

76075 

91230 

48 

13 

73863 

92252 

74999 

91746 

76093 

91221 

47 

14 

73882 

92244 

75017 

91738 

76111 

91212 

46 

15 

73901 

92235 

75036 

91729 

76129 

,91203 

45 

16 

73921 

92227 

75054 

91720 

76146 

91194 

44 

17 

73940 

92219 

75073 

91712 

76164 

91185 

43 

18 

73959 

92211 

75091 

91703 

76182 

91176 

42 

19 

73978 

92202 

75110 

91695 

76200 

91167 

41 

20 

9.73997 

9.92194 

9.75128 

9.  91  f  86 

9.76218 

9.91158 

40 

21 

74017 

92186 

75147 

91677 

76236 

91149 

39 

22 

74036 

92177 

75165 

91669 

76253 

91141 

38 

23 

74055 

92169 

75184 

91660 

76271 

91132 

37 

24 

74074 

92161 

75202 

91651 

76289 

91123 

36 

25 

74093 

92152 

75221 

91643 

76307 

91114 

35 

26 

74113 

92144 

75239 

91684 

76324 

91105 

34 

27 

74132 

92136 

75258 

91625 

76342 

91096 

33 

28 

74151 

92127 

75276 

91617 

76360 

91087 

32 

29 

74170 

92119 

75294 

91608 

76378 

91078 

31 

30 

9.74189 

9.92111 

9.75313 

9.91599 

9.76395 

9.91069 

30 

31 

74208 

92102 

75331 

91591 

76413 

91060 

29 

32 

74227 

92094 

75350 

91582 

76431 

91051 

28 

33 

74246 

92086 

75368 

91573 

76448 

91042 

27 

34 

74265 

92077 

75386 

91565 

76466 

91033 

26 

35 

74284 

92069 

75405 

91556 

76484 

91023 

25 

36 

74303 

92060 

75423 

91547 

76501 

91014 

24 

37 

74322 

92052 

75441 

91538 

76519 

91005 

23 

38 

74341 

92044 

75459 

91530 

76537 

90996 

22 

39 

74360 

92035 

75478 

91521 

76554 

90987 

21 

40 

9.74379 

9.92027 

9.75496 

9.91512 

9.76572 

9.90978 

20 

41 

74398 

92018 

75514 

91504 

76590 

90969 

19 

42 

74417 

92010 

75533 

91495 

76607 

90960 

18 

43 

74436 

92002 

75551 

91486 

76625 

90951 

17 

44 

74455 

91993 

75569 

91477 

76642 

90942 

16 

45 

74474 

91985 

75587 

91469 

76660 

90933 

15 

46 

74493 

91976 

75605 

91460 

76677 

90924 

14 

47 

74512 

91968 

75624 

91451 

76695 

90915 

13 

48 

74531 

91959 

75642 

91442 

76712 

90906 

12 

49 

74549 

91951 

75660 

91433 

76730 

90896 

11 

50 

9.74568 

9.01942 

9.75678 

9.91425 

9.76747 

9.90887 

10 

51 

74587 

91934 

75696 

91416 

76765 

90878 

9 

52 

74606 

91925 

75714 

91407 

76782 

90869 

8 

53 

74625 

91917 

75733 

91398 

76800 

90860 

54 

74644 

91908 

75751 

91389 

76817 

90851 

6 

55 

74662 

91900 

75769 

91381 

76835 

90842 

5 

56 

74681 

91891 

75787 

91372 

76852 

90832 

4 

57 

74700 

91883 

75805 

91363 

76870 

90823 

3 

58 

74719 

91874 

75823 

91354 

76887 

90814 

2 

59 

74737 

91866 

75841 

91345 

76904 

90805 

1 

60 

74756 

91857 

75859 

91336 

76922 

90796 

0 

f 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

J 

66° 

65° 

64° 

301 


TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


f 

36° 

37° 

38° 

i 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.76922 

9.90796 

9.77946 

9.90235 

9.78934 

9.89653 

GO 

1 

76939 

90787 

77963 

90225 

78950 

89643 

59 

2 

76957 

90777 

77980 

90216 

78967 

89633 

58 

3 

76974 

90768 

77997 

90206 

78983 

89624 

57 

4 

76991 

90759 

78013 

90197 

78999 

89614 

50 

5 

77009 

90750 

78030 

90187 

79015 

89604 

55 

6 

77026 

90741 

78047 

90178 

79031 

89594 

54 

7 

77043 

90731 

78063 

90168 

79047 

89584 

53 

8 

77061 

90722 

78080 

90159 

79063 

89574 

52 

9 

77078 

90713 

78097 

90149 

79079 

89564 

51 

10 

9.77095 

9.90704 

9.78113 

9.90139 

9.79095 

9.89554 

50 

11 

77112 

90694 

78130 

90130 

79111 

89544 

4!) 

12 

77130 

90685 

78147 

90120 

79128 

89534 

48 

13 

77147 

90676 

78163 

90111 

79144 

89524 

47 

14 

77164 

90667 

78180 

90101 

79160 

89514 

46 

15 

77181 

90657 

78197 

90091 

79176 

89504 

45 

16 

77199 

90648 

78213 

90082 

79192 

89495 

44 

17 

77216 

90639 

78230 

90072 

79208 

89485 

43 

18 

77233 

90630 

78246 

90063 

79224 

89475 

42 

19 

77250 

90620 

78263 

90053 

79240 

89465 

41 

20 

9.77268 

9.90611 

9.7S280 

9.90043 

9.79256 

9.89455 

40 

21 

77285 

90602 

78296 

90034 

79272 

89445 

39 

22 

77302 

90592 

78313 

90024 

79288 

89435 

38 

23 

77319 

90583 

78329 

90014 

79304 

89425 

37 

24 

77336 

90574 

7834G 

90005 

79319 

89415 

36 

25 

77353 

90565 

78362 

89995 

79335 

89405 

35 

26 

77370 

90555 

78379 

89985 

79351 

89395 

34 

27 

77387 

90546 

78395 

89976 

79367 

89385 

33 

28 

77405 

90537 

78412 

89966 

79383 

89375 

32 

29 

77422 

90527 

78428 

89956 

79399 

89364 

31 

30 

9.77439 

9.90518 

9.78445 

9.89947 

9.79415 

9.89354 

30 

31 

774E5 

90509 

78461 

89937 

79431 

89344 

29 

32 

77473 

90499 

78178 

89927 

79447 

89334 

28 

33 

77490 

90490 

78494 

89918 

79463 

89324 

27 

34 

77507 

90480 

78510 

89908 

79478 

89314 

26 

35 

77524 

90471 

785-27 

89898 

79494 

89304 

25 

36 

77541 

90462 

78543 

89888 

79510 

89291 

24 

37 

77553 

90452 

78560 

89879 

79526 

89284 

23 

38 

77575 

90443 

78576 

89869 

79542 

89274 

22 

39 

77592 

90434 

78592 

89859 

79558 

89264 

21 

40 

9.77609 

9.90424 

9.78609 

9.89849 

9.79573 

9.89254 

20 

41 

77626 

90415 

78625 

89840 

79589 

89244 

19 

42 

77643 

90405 

78642 

89830 

79605 

89233 

18 

43 

77060 

90396 

78658 

898-20 

78621 

89223 

17 

44 

77677 

90386 

78674 

89810 

79636 

89213 

16 

45 

77694 

90377 

78691 

89801 

79652 

89203 

15 

46 

77711 

9036S 

78707 

89791 

79668 

89193 

14 

47 

77728 

90358 

78723 

89781 

79684 

891  S3 

13 

48 

77744 

90349 

78739 

89771 

79699 

89173 

12 

49 

77761 

90339 

78756 

89761 

79715 

89162 

11 

50 

9.77778 

9.90330 

9.78772 

9.89752 

9.79731 

9.89152 

10 

51 

77795 

90320 

78788 

89742 

79746 

89146 

<) 

52 

77812 

90311 

78805 

89732 

79762 

89132 

8 

53 

77829 

90301 

78821 

89722 

79778 

89122 

7 

54 

77846 

90292 

78837 

89712 

79793 

89112 

6 

55 

77862 

90282 

78853 

89702 

79809 

89101 

5 

56 

77879 

90273 

78869 

89693 

79S25 

89091 

4 

57 

77896 

90263 

78886 

89683 

79840 

89081 

3 

58 

77913 

90254 

78902 

89G73 

79856 

89071 

2 

59 

77930 

90244 

78918 

89663 

79872 

890(50 

1 

60 

77946 

90235 

78934 

89653 

79887 

89050 

0 

~ 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

f 

63° 

52° 

51° 

TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


39° 

40° 

41° 

/ 

Sine 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

0 

9.79887 

9.89050 

9.80807 

9.88425 

9.81694 

9.87778 

60 

1 

79903 

89040 

80822 

88415 

81709 

87767 

59 

2 

79918 

89030 

80837 

88404 

81723 

87756 

58 

3 

79934 

89020 

80852 

88394 

81738 

87745 

57 

4 

79950 

89009 

80867 

88383 

81752 

87734 

56 

5 

79965 

88999 

80882 

88372 

81767 

87723 

55 

6 

79981 

88989 

80897 

88362 

81781 

87712 

54 

79996 

88978 

80912 

88351 

81796 

87701 

53- 

8 

80012 

88968 

80927 

88340 

81810 

87690 

52 

9 

80027 

88958 

80942 

88330 

81825 

87679 

51 

10 

9.80043 

9.88948 

9.80957 

9.88319 

9.81839 

9.87668 

60 

11 

80058 

88937 

80972 

88308 

81854 

87657 

49 

12 

80074 

88927 

80987 

88298 

81868 

87646 

48 

13 

80089 

88917 

81002 

88287 

81882 

87635 

47 

14 

80105 

88906 

81017 

88276 

81897 

87624 

46 

15 

80120 

88896 

81032 

88266 

81911 

87613 

45 

16 

80136 

88886 

81047 

88255 

81926 

87601 

44 

17 

80151 

88875 

81061 

88244 

81940 

87590 

43 

18 

80166 

88865 

81076 

88234 

81955 

87579 

42 

19 

80182 

88855 

81091 

88223 

81969 

87568 

41 

20 

9.80197 

9.88844 

9.81106 

9.88212 

9.81983 

9.87557 

40 

21 

80213 

88834 

81121 

88201 

81998 

87546 

39 

22 

80228 

88824 

81136 

88191 

82012 

87535 

38 

23 

80244 

88813 

81151   - 

88180 

8~>026 

87524 

37 

24 

80259 

8S803 

81166 

88169 

82041 

87513 

36 

25 

80274 

88793 

81180 

88158 

820.-5 

87501 

35 

26 

80290 

88782 

81195 

88148 

82069 

87490 

34 

27 

80305 

88772 

81210 

88137 

82084 

•87479 

33 

28 

80320 

88761 

81225 

82098 

87468 

32 

29 

80336 

88751 

81240 

88115 

82112 

87457 

31 

30 

9.80351 

9.88741 

9.81254 

9.88105 

9.82126 

9.87446 

30 

31 

80366 

88730 

81269 

88094 

82141 

87434 

29 

32 

80382 

88720 

81284 

88083 

82155 

87423 

28 

33 

80397 

88709 

81299 

88072 

82169 

87412 

27 

34 

80412 

88699 

81314 

88061 

82184 

87401 

26 

35 

80428 

88688 

81328 

88051 

82198 

87390 

25 

36 

80443 

88678 

81343 

88040 

82212 

87378 

24 

37 

80458 

88668 

81358 

88029 

82226 

67367 

23 

38 

80473 

88657 

81372 

88018 

82240 

87356 

22 

39 

80489 

88647 

81387 

88007 

82255 

87345 

21 

40 

9.80504 

9.88636 

9.81402 

9.87996 

9.82269 

9.87334 

20 

41 

80519 

88626 

81417 

87985 

82283 

87322 

19 

42 

80534 

88615 

81431 

87975 

82297 

87311 

18 

43 

80550 

88605 

81446 

87964 

82311 

87300 

17 

44 

80565 

88594 

81461 

87953 

8-J326 

87288 

16 

45 

80580 

88584 

81475 

87942 

82340 

87277 

15 

46 

80595 

8^573 

81490 

87931 

87266 

14 

47 

80610 

88563 

81505 

87920 

82368 

87255 

13 

48 

80625 

88552 

81519 

87909 

82382 

87243 

12 

49 

80641 

88542 

81534 

87898 

82396 

87232 

11 

50 

9.80656 

9.88531 

9.81549 

9.87887 

9.82410 

9.87221 

10 

51 

80671 

88521 

81563 

87877 

82424 

87209 

9 

52 

8(1686 

88510 

81578 

87866 

82439 

87198 

8 

53 

80701 

88499 

81592 

87855 

824f3 

87187 

7 

54 

80716 

88489 

81607 

87844 

82467 

87175 

6 

55 

80731 

88478 

81622 

87833 

82481 

87164 

5 

56 

80746 

88468 

81636 

87822 

82495 

87153 

4 

57 

80762 

88457 

81651 

87811 

82509 

87141 

3 

58 

80777 

88447 

81665 

87800 

82523 

87130 

2 

59 

80792 

88436 

81680 

87789 

82537 

87119 

1 

60 

80807 

88425 

81694 

87778 

82551 

87107 

0 

, 

Cosine 

Sine 

Cosine 

Sine 

Cosine 

Sine 

, 

50° 

49° 

48° 

303 


TABLE  XX.— LOGARITHMIC   SINES  AND  COSINES. 


42°    ' 


48° 


440 


Sine        Cosine 


Sine 


Cosine 


Sine 


Cosine 


0 

1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 


37 


59 


82579 
82593 
82607 


82635 
82649 


82677 

9.82691 
82705 
82719 
82733 
82747 
82r61 
82775 
82788 


82816 

9.82830 

S2844 


82913 


9.82968 
82982 


83010 


83051 
83065 
83078 


9.83106 
83120 
83133 
83147 
83161 
83174 


83215 
83229 


83270 
83283 
83297 
83310 
83324 


9.87107 
87096 
87085 
87073 
87062 
87050 


87028 
87016 
87005 


86970 


86913 


86867 
86855 
86844 
86832 


86786 
86775 

9.86763 
86752 
86740 
86728 
86717 
86705 


86670 


9.86647 
86635 


86612 


86577 
86565 
86554 
86542 

9.86530 
86518 
86507 
86495 
86483 
86472 


83351 

83365    86425 

83378    86413 


83405 
83419 
83432 
83446 


83540 
83554 
83567 


83621 
83634 


83674 


83701 
83715 
83728 
83741 
83755 


9.83781 
83795 


83821 


83874 


83927 


83967 


84006 
84020 
84033 

9.84046 
84059 
84072 
84085 
84098 
84112 
84125 
84138 
84151 
84164 
84177 


9.86413 
86401 


86377 


86354 
86342 


86318 


83478 
83486 
83500 

9.83513   9.86295 


86271 
86259 
86247 
86235 


86211 


9.86176 
86164 
86152 
86140 
86128 
86116 
86104 
86092 


9.86056 


85984 
85972 


85948 


85924 
85912 
85900 


85876 


85851 


85827 

9.85815 
85803 
85791 
85779 
85766 
85754 
85742 
85730 
85718 
85706 


9.84177 
84190 


84216 
84229 
84242 
84255 


84295 

9.84308 
84321 
84334 
84347 


84373 
84385 


84411 
84424 

9.84437 
84450 
84463 
84476 
84489 
84502 
84515 
84528 
84540 
84553 

9.84566 
84579 
84592 
84605 
84618 
84630 
84643 
84656 


9.84694 
84707 
84720 
84733 
84745 
84758 
84771 
84784 
84796 
84809 

9.84822 
84835 
84847 
84860 
84873 
84885 


84911 
84923 
84936 
84949 


85681 


85657 
85645 
85G32 


85608 


85583 

1.85571 
85559 
85547 
85534 
85522 
85510 
85497 
85485 
85473 
85460 

1.85448 
85436 
85423 
85411 
85399 
85386 
85874 
85361 
85349 
85337 


85312 
85299 


85274 


85250 
85237 
85225 
85212 

9.85200 
85187 
85175 


85150 
85137 
85125 
85112 
85100 
85087 

9.85074 


85049 
85037 
85024 
85012 


84974 
84961 
84949 


Cosine        Sine 
47° 


Cosine 


Sine 


Cosine 


Sine 


46a 

304 


45° 


TABLE  XXI.— LOG.  TANGENTS  AND  COTANGENTS. 


0° 

1° 

2° 

f 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

—  oo 

00 

8.24192 

11.75808 

8.54308 

11.45692 

60 

1 

6.46373 

13.53627 

24910 

75090 

54669 

45331 

59 

2 

76476 

23524 

25616 

74384 

55027 

44973 

58 

3 

94085 

05915 

26312 

73688 

55382 

44618 

57 

4 

7.06579 

12.93421 

26996 

73004 

55734 

44266 

56 

5 

16270 

83730 

27669 

72331 

56083 

43917 

55 

6 

24188 

75812 

28332 

71668 

56429 

43571 

54 

7 

30882 

69118 

28986 

71014 

56773 

43227 

53 

8 

36682 

63318 

29629 

70371 

57114 

42886 

52 

9 

41797 

58203 

30263 

69737 

57452 

42548 

61 

10 

7.46373 

12.53627 

8.30888 

11.69112 

8.57788 

11.42212 

50 

11 

50512 

49488 

31505 

68495 

58121 

41879 

49 

12 

54291 

45709 

32112 

67888 

58451 

41549 

48 

13 

57767 

42233 

32711 

67289 

58779 

41221 

47 

14 

60986 

39014 

33302 

66698 

59105 

40895 

46 

15 

63982 

36018 

33886 

66114 

59428 

40572 

45 

16 

66785 

33215 

34461 

65539 

59749 

40251 

44 

17 

69418 

30582 

35029 

64971 

60068 

39932 

43 

18 

71900 

28100 

35590 

64410 

60384 

39616 

42 

19 

74248 

25752 

36143 

63857 

60698 

89302 

41 

20 

7.76476 

12.23524 

8.36689 

11.63311 

8.61009 

11.38991 

40 

21 

78595 

21405 

37229 

62771 

61319 

36681 

39 

22 

80615 

19385 

37762 

62238 

61626 

38374 

38 

23 

82546 

17454 

38289 

61711 

61931 

38069 

37 

24 

84394 

15606 

38809 

61191 

62234 

37766 

36 

25 

86167 

13833 

39323 

60677 

62535 

37465 

35 

26 

87871 

12129 

39832 

60168 

62834 

37166 

34 

27 

89510 

10490 

40334 

59666 

63131 

36869 

33 

28 

91089 

08911 

40830 

59170 

63426 

36574 

32 

29 

92613 

07387 

41321 

58679 

63718 

36282 

31 

30 

7.94086 

12.05914 

8.41807 

11.58193 

8.64009 

11.35991 

30 

31 

95510 

04490 

42287 

57713 

64298 

85702 

29 

32 

96889 

03111 

42762 

57238 

64585 

35415 

28 

33 

98225 

01775 

43232 

56768 

64870 

35130 

27 

34 

99522 

00478 

43696 

56304 

65154 

34846 

26 

35 

8.00781 

11.99219 

44156 

55844 

65435 

34565 

25 

36 

02004 

97996 

44611 

55389 

65715 

34285 

24 

37 

03194 

96806 

45061 

54939 

65993 

34007 

23 

38 

04353 

95647 

45507 

54493 

66269 

33731 

22 

39 

05481 

94519 

45948 

54052 

66543 

33457 

21 

40 

8.06581 

11.93419 

8.46385 

11.53615 

8.66816 

11.33134 

20 

41 

07653 

92347 

46817 

53183 

67087 

32913 

19 

42 

08700 

91300 

47245 

52755 

67356 

32644 

18 

43 

09722 

90278 

47669 

52331 

67624 

32376 

17 

44 

10720 

89280 

48089 

51911 

67890 

32110 

16 

45 

11696 

88304 

48505 

51495 

68154 

31846 

15 

46 

12651 

87349 

48917 

51083 

68417 

31583 

14 

47 

13585 

86415 

49325 

50675 

68678 

31322 

13 

48 

14500 

85500 

49729 

50271 

68938 

31062 

12 

49 

15395 

84605 

50130 

49870 

69196 

30804 

11 

50 

8.16273 

11.83727 

8.50527 

11.49473 

8.69453 

11.30547 

10 

51 

17133 

82867 

50920 

49080 

69708 

30292 

9 

52 

17976 

82024 

51310 

48690 

69962 

30038 

8 

53 

18804 

81196 

51696 

48304 

70214 

29786 

7 

54 

19616 

80384 

52079 

47921 

70465 

29535 

6 

55 

20413 

79587 

52459 

47541 

70714 

29286 

5 

56 

21195 

78805 

52835 

47165 

70962 

29038 

4 

57 

21964 

78036 

53208 

46792 

71208 

28792 

3 

58 

22720 

77280 

53578 

46422 

71453 

28547 

2 

59 

23462 

76538 

53945 

46055 

71697 

28303 

1 

60 

24192 

75808 

54308 

45692 

71940 

28060    0 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

i 

89° 

88° 

87° 

305 


TABLE  XXI.— LOG.  TANGENTS   AND   COTANGENTS. 


/ 

3° 

4° 

5° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

8.71940 

11.28060 

8.84464 

11.15536 

8.94195 

11.05805 

60 

1 

72181 

27819 

84646 

15354 

94340 

05660 

59 

2 

72420 

27580 

84826 

15174 

94485 

05515 

58 

3 

72659 

27341 

85006 

14994 

94630 

05370 

57 

4 

72896 

27104 

85185 

14815 

94773 

05227 

56 

5 

73132 

26868 

85363 

14637 

94917 

05083 

55 

6 

73366 

26634 

85540 

14460 

95060 

04940 

54 

7 

73600 

26400 

85717 

14283 

95202 

04798 

53 

8 

73832 

26163 

85893 

14107 

95344 

04656 

52 

9 

74063 

25937 

86069 

13931 

95486 

04514 

51 

10 

8.74292 

11.25708 

8.86243 

11.13757 

8.95627 

11.04373 

50 

11 

745-21 

25479 

86417 

13583 

95767 

04233 

49 

12 

74748 

25252 

86591 

13409 

95908 

04092 

48 

13 

74974 

25026 

86763 

13-237 

96C47 

03953 

47 

14 

75199 

24801 

86935 

13065 

96187 

03813 

46 

15 

75423 

24577 

8H06 

12894 

96325 

03675 

45 

16 

75645 

24355 

87277 

12723 

96464 

03536 

44 

ir 

75867 

24133 

87447 

12553 

96G02 

03398 

43 

18 

76087 

23913 

87616 

12384 

96739 

03261 

42 

19 

76306 

23694 

877'85 

12215 

96877 

02123 

41 

20 

8.765-25 

11.23475 

8.87953 

11.12047 

8.97013 

11.02987 

40 

21 

76742 

23258 

88150 

11880 

97150 

0'2850 

39 

22 

76958 

23042 

88-287 

11713 

97285 

02715 

38 

23 

77173 

228-27 

88453 

11547 

97421 

02579 

37 

24 

77387 

22613 

88618 

11382 

97556 

0-2444 

36 

25 

77600 

22400 

88783 

11217 

97691 

.   02309 

35 

26 

77811 

22189 

88948 

11052 

97825 

02175 

34 

27 

78022 

21978 

89111 

10889 

97959 

02041 

33 

28 

78232 

21768 

89274 

10726 

9809-2 

01908 

32 

29 

78441 

21559 

89437 

10563 

98225 

01775 

31 

30 

8.78649 

11.21351 

8.89598 

11.1040-2 

8.98358 

11.01642 

80 

31 

78855 

21145 

89760 

10240 

98490 

01510 

29 

32 

79061 

20939 

89920 

10080 

986'22 

01378 

28 

83 

79'266 

20734 

90080 

09920 

98753 

01247 

27 

34 

79470 

20530 

90i>40 

09760 

98884 

01116 

26 

35 

79673 

203'27 

90399 

09601 

99015 

00985 

25 

36 

79875 

20125 

90557 

09443 

99145 

00855 

24 

37 

80076 

19924 

90715 

09-285 

99-275 

00725 

23 

38 

80277 

19723 

90872 

09128 

99405 

00595 

22 

39 

80476 

19524 

910-.'9 

08971 

99534 

00466 

21 

40 

8.80674 

11.19326 

8.91185 

11.08815 

8.99662 

11.00338 

20 

41 

80872 

19128 

91340 

08660 

99791 

00209 

19 

42 

81068 

18932 

91495 

08505 

99919 

00081 

18 

43 

81264 

18736 

91650 

08350 

9.00046 

10.90954 

17 

44 

81459 

18541 

91803 

08197 

00174 

99P'26 

16 

45 

81653 

18347 

91957 

08043 

00301 

99699 

15 

46 

81846 

18154 

9-2110 

07890 

00427 

99573 

14 

47 

82038 

17962 

92262 

07738 

00553 

99447 

18 

48 

82230 

17770 

9-2414 

07586 

00679 

998-21 

12 

49 

82420 

17580 

92565 

07435 

00805 

99195 

11 

50 

8.82610 

11.17390 

8.92716 

11.07284 

9.00930 

10.99070 

10 

51 

82799 

17201 

92866 

07134 

01055 

98945 

9 

52 

8-2987 

17013 

93016 

06984 

01179 

98821 

8 

53 

83175 

168-25 

93165 

06835 

01303 

98697 

7 

54 

83361 

16639 

93313 

06G87 

01427 

98573 

6 

55 

83547 

16453 

93462 

06538 

01550 

98450 

5 

56 

83732 

16268 

93609 

06391 

01673 

98327 

4 

57 

83916 

16084 

93756 

06244 

01796 

98204 

3 

58 

84100 

15900 

93903 

06097 

01918 

98082 

2 

59 

84282 

15718 

94049 

05951 

02040 

97960 

1 

60 

84464 

15536 

94195 

05805 

02162 

97838 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

86° 

85° 

84° 

TABLE  XXL— LOG.  TANGENTS   AND   COTANGENTS. 


/ 

6 

o 

7° 

8° 

; 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.02162 

10.97838 

9.08914 

10.91086 

9.14780 

10.85220 

60 

1 

02283 

97717 

09019 

90981 

14872 

85128 

59 

2 

02404 

97596 

09123 

90877 

14963 

85037 

58 

3 

025-J5 

97475 

09227 

90773 

15054 

84946 

57 

4 

02645 

97355 

09330 

90670 

15145 

84855 

56 

5 

02766 

97234 

09434 

90566 

15236 

84764 

55 

6 

02H85 

97115 

09537 

90463 

15327 

84673 

54 

7 

03005 

96995 

09640 

90360 

15417 

84583 

53 

8 

03124 

96876 

09742 

90258 

15508 

84492 

52 

9 

03242 

96758 

09845 

90155 

15598 

84402 

51 

10 

9.03361 

10.96639 

9.09947 

10.90053 

9.15688 

10.84312 

50 

11 

03479 

96521 

10049 

89951 

15777 

842:23 

49 

12 

03597 

96403 

10150 

89850 

15867 

84133 

48 

13 

03714 

96286 

10252 

89748 

15956 

84044 

47 

14 

03832 

96168 

10353 

89647 

16046 

83954 

46 

15 

03948 

96052 

10454 

89546 

16)35 

83865 

45 

16 

04065 

95935 

10555 

89445 

16224 

83776 

44 

17 

04181 

95819 

10656 

89344 

16812 

83688 

43 

18 

04-297 

95703 

10756 

89244 

16401 

83599 

42 

19 

04413 

95587 

10856 

89144 

16489 

83511 

41 

20 

9.045-28 

10.95472 

9.10956 

10.89044 

9.16577 

10.83423 

40 

21 

04643 

95357 

11056 

88944 

16665 

83335 

39 

22 

04758 

95242 

11155 

88845 

16753 

83247 

38 

23 

04873 

95127 

11254 

88746 

16841 

83159 

37 

24 

04987 

95013 

11353 

88647 

16928 

83072 

86 

25 

05101 

94899 

11452 

88548 

17016 

82984 

35 

26 

05214 

94786 

11551 

88449 

17103 

82897 

34 

27 

053-28 

94672 

11649 

88351 

17190 

82810 

33 

28 

05441 

94559 

11747 

88253 

17277 

82723 

32 

29 

05553 

94447 

11845 

88155 

17363 

82637 

31 

30 

9.05666 

10.94334 

9.11943 

10.88057 

9;  17450 

10.82550 

30 

31 

05778 

94222 

12040 

87960 

17536 

82464 

29 

32 

05890 

94110 

12138 

87862 

17622 

82378 

28 

33 

06002 

93998 

12235 

87765 

17708 

82292 

27 

34 

06113 

93887 

12332 

87668 

17794 

82206 

26 

35 

06224 

93776 

12428 

87572 

17880 

82120 

25 

36 

06335 

93665 

12525 

87475 

17965 

82035 

24 

37 

06445 

93555 

12621 

87379 

18051 

81949 

23 

38 

06556 

93444 

12717 

87283 

18136 

81864 

22 

39 

00666 

93334 

12813 

87187 

18221 

81779 

21 

40 

9.06775 

10.93225 

9.12909 

10.87091 

9.18306 

10.81694 

20 

41 

06885 

93115 

13004 

86996 

18391 

81609 

19 

42 

06994 

93006 

13099 

86901 

18475 

81525 

18 

43 

07103 

92897 

13194 

86806 

18560 

81440 

17 

44 

07-211 

92789 

13289 

86711 

18644 

81856 

16 

45 

07320 

92680 

13384 

86616 

187'28 

81272 

15 

46 

07428 

92572 

13178 

86522 

18812 

81188 

14 

47 

07536 

9:2464 

13573 

86427 

18896 

81104 

13 

48 

07643 

9-2357 

13667 

86338 

18979 

81021 

12 

49 

07751 

92249 

13761 

86239 

19063 

80937 

11 

50 

9.07858 

10.92142 

9.13854 

10.86146 

9.19146 

10.80854 

10 

51 

07964 

92036 

13948 

86052 

19-229 

80771 

9 

52 

08071 

91929 

14041 

85959 

19312 

80688 

8 

53 

08177 

91P'23 

14134 

85866 

19395 

80605 

7 

54 

08288 

91717 

14227 

85773 

19478 

80522 

6 

55 

08389 

91(511 

14320 

85680 

19561 

80439 

5 

56 

08495 

91505 

14412 

855S8 

19643 

80357 

4 

57 

08600 

91400 

14504 

85496 

19725 

80275 

3 

58 

08705 

91295 

14597 

85403 

19807 

80193 

2 

59 

08810 

91190 

14688 

85312 

19889 

80111 

1 

60 

08914 

91086 

14780 

85220 

19971 

800-29 

0 

t 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

83° 

82° 

81° 

307 


-LOG.  TANGENTS  AND  COTANGENTS. 


• 
1 

9° 

10° 

11° 

1 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.19971 

10.80029 

9.24632 

10.75368 

9.28865 

10.71135 

60 

1 

20053 

79947 

24706 

75294 

28933 

71067 

59 

2 

20134 

79866 

24779 

75221 

29000 

71000 

58 

3 

20216 

79784 

24853 

75147 

29067 

70933 

57 

4 

20297 

79703 

24926 

75074 

29134 

70866 

56 

5 

20378 

79622 

25000 

75000 

29201 

70799 

55 

6 

20459 

79541 

25073 

74927 

29268 

70732 

54 

7 

20540 

79460 

25146 

74854 

29335 

70665" 

53 

8 

20621 

79379 

25219 

74781 

29402 

70598 

62 

9 

20701 

79299 

25292 

74708 

29468 

70532 

51 

10 

9.20782 

10.79218 

9.25365 

10.74635 

9.29535 

10.70465 

50 

11 

20862 

79138 

25437 

74563 

29601 

70399 

49 

12 

20942 

79058 

25510 

74490 

29668 

70332 

48 

13 

21022 

78978 

25582 

74418 

29734 

70266 

47 

14 

21102 

78898 

25655 

74345 

29800 

70200 

46 

15 

21182 

78818 

25727 

74273 

29866 

70134 

45 

16 

21261 

78739 

25799 

74201 

29932 

70068 

44 

17 

21341 

78659 

25871 

74129 

29998 

70002 

43 

18 

21420 

78580 

25943 

74057 

30064 

69936 

42 

19 

21499 

78501 

26015 

73985 

30130 

69870  ' 

41 

80 

9.21578 

10.78422 

9.26086 

10.73914 

9.30195 

10.69805 

40 

21 

21657 

78343 

26158 

73842 

30261 

69739 

39 

22 

21736 

78264 

26229 

73771 

30326 

6967  4 

38 

23 

21814 

78186 

26301 

73699 

30391 

G9609 

87 

24 

21893 

78107 

26372 

73628 

30457 

69543 

36 

25 

21971 

78029 

26443 

73557 

30522 

69478 

35 

26 

22049 

77951 

26514 

73486 

30587 

69413 

34 

27 

22127 

77873 

26585 

73415 

30652 

69348 

33 

28 

22205 

77795 

26655 

73345 

30717 

69283 

32 

29 

22283 

77717 

26726 

73274 

30782 

69218 

31 

30 

9.22361 

10.77639 

9.26797 

10.73203 

9.30846 

10.69154 

30 

31 

22438 

77562 

26S67 

73133 

30911 

69089 

29 

32 

22516 

77484 

26937 

73063 

30975 

69025 

28 

33 

22598 

77407 

27008 

72992 

31040 

68960 

27 

34 

22670 

77330 

27078 

72922 

31104 

68896 

26 

35 

22747 

77253 

27148 

72852 

31168 

68832 

25 

36 

22824 

77176 

27218 

72782 

31233 

68767 

24 

87 

22901 

77099 

27288 

72712 

31297 

68703 

23 

88 

22977 

77023 

27357 

72648 

31361 

68639 

22 

39 

23054 

76946 

27427 

72573 

31425 

68575 

21 

40 

9.23130 

10.76870 

9.27496 

10.72504 

9.31489 

10.68511 

20 

41 

28206 

76794 

27566 

72434 

31552 

68448 

19 

42 

23283 

76717 

27635 

72365 

31616 

68384 

18 

43 

23359 

76641 

27704 

72296 

31679 

68321 

17 

44 

23435 

76565 

27773 

72227 

31743 

68257 

16 

45 

23510 

76490 

27842 

72158 

81806 

68194 

15 

46 

23586 

76414 

27911 

72089 

31870 

68130 

14 

47 

23661 

76339 

27980 

72020 

31933 

68067 

13 

48 

23787 

76263 

28049 

71951 

31996 

68004 

12 

49 

23312 

76188 

28117 

71883 

32059 

67941 

11 

50 

9.23887 

10.76113 

9.28186 

10.71814 

9.32122 

10.67878 

10 

51 

23962 

76038 

28254 

71746 

32185 

67815 

9 

52 

24037 

75963 

28323 

71677 

32248 

67752 

8 

53 

24113 

75888 

28391 

71609 

3:2311 

67689 

7 

54 

24186 

75814 

28459 

71541 

32373 

67627 

6 

55 

24261 

75739 

28527 

71473 

32436 

67564 

5 

56 

24335 

75665 

28595 

71405 

32498 

67502 

4 

67 

24410 

75590 

28662 

71338 

32561 

67439 

3 

58 

24484 

75516 

28730 

71270 

32623 

67377 

2 

59 

24558 

75442 

28798 

71202 

32685 

67315 

1 

60 

24632 

75368 

28865 

71135 

32747 

67253 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

80* 

79° 

78° 

TABLE  XXI.— LOG.  TANGENTS  AND   COTANGENTS. 


Jj 

12° 

13° 

14° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.32747 

10.67253 

9.36336 

10.63664 

9.39677 

10.60323 

60 

1 

32810 

67190 

36394 

63606 

39731 

60269 

59 

2 

32872 

67128 

36452 

63548 

39785 

60215 

58 

3 

32933 

67067 

36509 

63491 

39838 

60162 

57 

4 

32995 

67005 

36566 

63434 

89892 

60108 

56 

5 

33057 

66943 

36624 

63376 

39945 

60055 

55 

g 

33119 

66881 

36681 

63319 

39999 

60001 

54 

7 

83180 

66820 

36738 

63262 

40052 

59948 

53 

8 

S3242 

66758 

36795 

63205' 

40106 

59894 

52 

9 

83303 

66697 

36852 

63148 

40159 

59841 

61 

10 

9.33365 

10.66635 

9.36909 

10.63091 

9.40212 

10.59788 

50 

11 

33426 

66574 

36966 

63034 

40266 

59734 

49 

12 

33487 

66513 

37023 

62977 

40319 

59681 

48 

13 

33548 

66452 

37080 

62920 

40372 

59628 

47 

14 

33609 

66391 

37137 

628G3 

40425 

59575 

46 

15 

33670 

66330 

87193 

62807 

40478 

59522 

45 

16 

33731 

66269 

37250 

62750 

40531 

59469 

44 

17 

33792 

66208 

37306 

62G94 

40584 

59416 

43 

18 

83853 

66147 

37363 

62637 

40636 

59364 

42 

19 

33913 

66087 

37419 

62581 

40689 

59311 

41 

20 

9.33974 

10.66026 

9.37476 

10.62524 

9.40742 

10.59258 

40 

21 

34034 

65966 

37532 

62468 

40795 

69205 

39 

22 

34095 

65905 

37588 

62412 

40847 

59153 

38 

23 

84155 

65845 

37644 

62356 

40900 

59100 

37 

24 

34215 

65785 

37700 

62300 

40952 

59048 

36 

25 

84276 

65724 

37756 

62244 

41005 

58995 

35 

26 

34336 

65664 

87812 

62188 

41057 

58943 

34 

27 

34396 

65604 

37868 

62132 

41109 

58891 

33 

28 

34456 

65544 

37924 

62076 

41161 

58839 

32 

29 

34516 

65484 

37980 

62020 

41214 

58786 

31 

30 

9.34576 

10.65424 

9.38035 

10.61965 

9.41266 

10.58734 

30 

31 

34635 

65365 

38091 

61909 

41318 

58682 

29 

32 

34695 

65305 

38147 

61853 

41370 

58630 

28 

33 

34755 

65245 

38202 

61798 

41422 

58578 

27 

34 

34814 

65186 

38257 

61743 

41474 

58526 

26 

35 

34874 

65126 

38313 

61687 

41526 

58474 

25 

36 

34933 

65067 

38368 

61632 

41578 

58422 

24 

37 

34992 

65008 

38423 

61577 

41629 

58371 

23 

38 

35051 

64949 

38479 

61521 

41681 

5&319 

22 

39 

35111 

64889 

38534 

61466 

41733 

58267 

21 

40 

9.35170 

10.64830 

9.38589 

10.61411 

9.41784 

10.58216 

20 

41 

35229 

64771 

38644 

61356 

41836 

58164 

19 

42 

35288 

64712 

38699 

61301 

41887 

58113 

18 

43 

35347 

64653 

38754 

61246 

41939 

58061 

17 

44 

35405 

64595 

38808 

61192 

41990 

58010 

16 

45 

35464 

64536 

38863 

61137 

42041 

57959 

15 

46 

35523 

64477 

38918 

61082 

42093 

57907 

14 

47 

35581 

64419 

38972 

61028 

42144 

57856 

13 

48 

85640 

64360 

39027 

60973 

42195 

57805 

12 

49 

3569S 

64302 

39082 

60918 

42246 

57754 

11 

50 

9.35757 

10.64243 

9.39136 

10.60864 

9.42297 

10.57703 

10 

51 

35815 

64185 

39190 

60810 

42348 

57652 

9 

52 

35873 

64127 

39245 

60755 

42399 

57601 

8 

53 

35931 

64069 

39299 

60701 

42450 

57550 

7 

54 

35989 

64011 

39353 

60647 

42501 

57499 

6 

55 

36047 

63953 

39407 

60593 

42552 

57448 

5 

56 

36105 

63895 

89461 

60539 

42603 

57397 

4 

57 

36163 

63837 

39515 

60485 

42653 

57347 

3 

58 

36221 

63779 

39569 

60431 

42704 

57296 

2 

59 

36279 

63721 

39623 

60377 

42755 

57245 

1 

60 

36336 

63664 

39677 

60323 

42805 

57195 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

j 

77° 

76° 

75° 

TABLE  XXI.— LOG.  TANGENTS  AND   COTANGENTS. 


f 

15° 

16° 

17° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.42805 

10.57195 

9.45750 

10.54250 

9.48534 

10.51466 

60 

1 

42856 

57144 

45797 

54203 

48579 

51421 

59 

2 

42906 

57094 

45845 

54155 

48624 

51376 

58 

3 

42957 

57043 

45892 

54108 

48669 

51331 

57 

4 

43007 

56993 

45940 

54060 

48714 

51286 

56 

5 

43057 

56943 

45987 

54013 

48759 

51241 

55 

6 

43108 

56892 

46035 

53965 

48804 

51196 

54 

7 

43158 

56842 

.46082 

53918 

48849 

51151 

S3 

8 

43208 

56792 

46130 

53870 

48894 

51106 

53 

9 

43258 

56742 

46177 

53823 

48939 

51061 

51 

10 

9.43308 

10.56692 

9.46224 

10.53776 

9.48984 

10.51016 

50 

11 

43358 

56642 

46271 

53729 

49029 

50971 

49 

12 

43408 

5659.2 

46319 

53681 

49073 

50927 

48 

18 

43458 

56542 

46366 

53634 

'49118 

50882 

47 

14 

43508 

56492 

46413 

53587 

49163 

50837 

46 

15 

43558 

56442 

46460 

53540 

49207 

50793 

45 

16 

43607 

56393 

46507 

53493 

49252 

50748 

44 

17 

43657 

56343 

46554 

53446 

49296 

50704 

43 

18 

43707 

56293 

46601 

53399 

49341 

50659 

42 

19 

43756 

56244 

46648 

53352 

49385 

50615 

41 

20 

9.43806 

10.56194 

9.46694 

10.53306 

9.49430 

10.50570 

40 

21 

43855 

56145 

46741 

53259 

49474 

50526 

39 

22 

43905 

56095 

46788 

53212 

49519 

50481 

38 

H 

43954 

56046 

46835 

53165 

49563 

50437 

37 

24 

44004 

55996 

46881 

53119 

49607 

50393 

36 

25 

44053 

55947 

46928 

53072 

49652 

50348 

35 

26 

44102 

55898 

46975 

53025 

49696 

50304 

34 

27 

44151 

55849 

47021 

52979 

49740 

50260 

33 

28 

44201 

55799 

47068 

52932 

49784 

50216 

32 

29 

44250 

55750 

47114 

52886 

49828 

50172 

31 

30 

9.44299 

10.55701 

9.47160 

10.52840 

9.49872 

10.50128 

30 

31 

44348 

55652 

47207 

52793 

49916 

50084 

29 

32 

44397 

55603 

47253 

52747 

49960 

50040 

28 

33 

44446 

55554 

47299 

52701 

50004 

49996 

34 

44495 

55505 

47346 

52654 

50048 

49952 

26 

35 

44544 

55456 

47392 

52608 

50092 

49908 

25 

36 

44592 

55408 

47438 

52562 

50136 

49864 

24 

37 

44641 

55359 

'47484 

52516 

50180 

49820 

23 

38 

44690 

55310 

47530 

52470 

50223 

49777 

22 

39 

44738 

55262 

47576 

52424 

50267 

49733 

21 

40 

9.44787 

10.55213 

9.47622 

10.52378 

9.50311 

10.49689 

20 

41 

44836 

55164 

47668 

52332 

50355 

49645 

19 

42 

44884 

55116 

47714 

52286 

50398 

49602 

18 

43 

44933 

55067 

47760 

52240 

50442 

49558 

17 

44 

44981 

55019 

47806 

52194 

50485 

49515 

16 

45 

45029 

54971 

47852 

52148 

50529 

49471 

15 

46 

45078 

54922 

47897 

52103 

50572 

49428 

14 

47 

'45126 

54874 

47943 

52057 

50616 

49384 

13 

48 

45174 

54826 

47989 

52011 

50659 

49341 

12 

49 

45222 

54778 

48035 

51965 

50703 

49297 

11 

50 

9.45271 

10.54729 

9.48080 

10.51920 

9.50746 

10.49254 

10 

51 

45319 

54681 

48126 

51874 

50789 

49211 

9 

52 

45367 

54633 

48171 

51829 

50833 

49167 

8 

53 

45415 

54585 

48217 

51783 

56876 

49124 

7 

54 

45463 

54537 

48262 

51738 

50919 

49081 

6 

K 

45511 

54489 

48307 

51693 

50962 

49088 

5 

56 

45559 

54441 

48353 

51647 

51005 

48995 

4 

57 

45606 

54394 

48398 

51602 

51048 

48952 

3 

58 

45654 

54346 

48443 

51557 

51092 

48908 

2 

59 

457C2 

54298 

48489 

51511 

51135 

48H65 

1 

60 

45750 

54250 

48534 

51466 

51178 

48822 

0 

/ 

Co  tan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

74° 

73° 

72° 

, 

310 


TABLE  XXI.— LOG.  TANGENTS   AND   COTANGENTS. 


/ 

18° 

19° 

20° 

t 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.51178 

10.48822 

9.53697 

10.46303 

9.56107 

10.43893 

60 

1 

51221 

48779 

53738 

46262 

56146 

43854 

59 

2 

51264 

48736 

53779 

46221 

56185 

438.5 

58 

3 

51306 

48694 

53820 

46180 

56:224 

43776 

57 

4 

51  349 

48651 

53861 

46139 

56264 

43736 

56 

5 

51392 

48608 

53902 

46098 

56303 

43697 

55 

6 

51435 

48565 

53943 

46057 

56342 

43658 

54 

7 

51478 

48522 

53984 

46016 

56381 

43619 

53 

8 

51520 

48480 

540:25 

45975 

56420 

43580 

52 

9 

51563 

48437 

54065 

45935 

56459 

43541 

51 

10 

9.51606 

10.48394 

9.54106 

10.45894 

9.56498 

10.43502 

50 

11 

51648 

48352 

54147 

45&53 

56537 

43463 

49 

12 

51691 

48309 

54187 

45813 

56576 

43424 

48 

13 

51734 

48266 

54228 

45772 

56815 

43385 

47 

14 

51776 

48224 

54269 

45731 

56654 

43346 

46 

15 

51819 

48181 

54309 

45691 

56693 

43307 

45 

16 

51861 

48139 

54350 

45650 

56732 

43268 

44 

17 

51903 

48097 

54390 

45610 

56771 

432-29 

43 

18 

51946 

48054 

54431 

45569 

56810 

43190 

42 

19 

51988 

48012 

54471 

45529 

56849 

43151 

41 

20 

9.52031 

10.47969 

9.54512 

10.45488 

9.56887 

10.43113 

40 

21 

52073 

47927 

54552 

45448 

56926 

43074 

39 

22 

52115 

47885 

54593 

45407 

56965 

43035 

38 

23 

52157 

47843 

54633 

45367 

57004 

42996 

37 

24 

52200 

47800 

54673 

45327 

57042 

48958 

86 

25 

5224-2 

47758 

54714 

45286 

57081 

42919 

35 

26 

52284 

47716 

54754 

45246 

57120 

42880 

34 

27 

52326 

47674 

54794 

45206 

57158 

42842 

33 

28 

52368 

47632 

54835 

45165 

57197 

42803 

32 

29 

52410 

47590 

54875 

45125 

57235 

42765 

i 

30 

9.52452 

10.47548 

9.54915 

10.45085 

9.57274 

10.42726 

30 

31 

52494 

47506 

54955 

45045 

57312 

42688 

29 

32 

52536 

47464 

54995 

45005 

57351 

42649 

28 

33 

5-2578 

47422 

55035 

44965 

57389 

42611 

27 

34 

52620 

47380 

55075 

449-25 

57428 

42572 

26 

35 

52661 

47339 

55115 

44885 

57466 

42534 

25 

36 

52703 

47297 

55155 

44845 

57504 

42496 

24 

37 

52745 

47255 

55195 

44805 

57543 

42457 

23 

38 

5-2787 

47213 

55235 

44765 

57581 

42419 

22 

39 

5-2829 

47171 

55275 

44725 

57619 

42381 

21 

40 

9.5-2870 

10.47130 

9.55315 

10.44685 

9.57658 

10.42342 

20 

41 

52912 

47088 

55355 

44645 

57696 

4*304 

19 

42 

5-2953 

47047 

55395 

44605 

57734 

42266 

18 

43 

52995 

47005 

55434 

44566 

57772 

42:228 

17 

44 

53037 

46963 

55474 

445-26 

57810 

42190 

16 

45 

53078 

4G922 

55514 

44486 

57849 

42151 

15 

46 

53120 

46880 

55554 

44446 

57887 

42113 

14 

47 

53161 

46839 

55593 

44407 

57925 

42075 

13 

48 

53202 

46798 

55633 

44367 

57963 

42037 

12 

49 

53244 

46756 

55673 

44327 

5S001 

41999 

11 

50 

9.53285 

10.46715 

9.55712 

10.44288 

9.58039 

10.41961 

10 

51 

53327 

46673 

55752 

44248 

58077 

41923 

9 

52 

58868 

46632 

55791 

44209 

58115 

41885 

8 

53 

53409 

46591 

55831 

44169 

58153 

41847 

•;• 

54 

53450 

46550 

55870 

44130 

58191 

41809 

6 

55 

53492 

46508 

55910 

44090 

582:29 

41771 

5 

56 

53533 

46467 

55949 

44051 

58567 

41733 

4 

57 

53574 

46426 

55989 

44011 

58304 

41696 

3 

58 

53615 

46385 

56028 

43972 

5834:3 

41658 

2 

59 

53656 

46344 

56067 

43933 

58380 

41620 

1 

60 

53697 

46303 

56107 

43893 

58418 

41582 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

i 

71° 

70° 

69° 

311 


TABLE  XXI.— LOG.  TANGENTS  AND  COTANGENTS. 


t 

21° 

22° 

23° 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.58418 

10.41582 

9.60641 

10.39359 

9.62785 

10.37215 

60 

1 

58455 

41545 

60677 

39323 

62820 

37180 

59 

2 

58493 

41507 

60714 

39286 

62855 

37145 

58 

3 

58531 

41469 

60750 

39250 

62890 

37110 

57 

4 

58569 

41431 

60786 

39214 

62926 

3707'4 

56 

5 

58606 

41394 

60823 

39177 

62961 

37039 

55 

6 

58644 

41356 

60859 

39141 

62996 

37004 

54 

7 

58681 

41319 

60895 

39105 

63031 

36969 

53 

8 

58719 

41281 

60931 

39069 

63066 

36934 

52 

9 

58757 

41243 

60967 

39033 

63101 

36899 

51 

10 

9.58794 

10.41206 

9.61004 

10.38996 

9.63135 

10.36B65 

50 

11 

58832 

41168 

61040 

38960 

63170 

36830 

49 

12 

58869 

41131 

61076 

38924 

63205 

36795 

48 

13 

58907 

41093 

61112 

38888 

63240 

36760 

47 

14 

58944 

41056 

61148 

38852 

63275 

36725 

46 

15 

58981 

41019 

61184 

38816 

63310 

30690 

45 

16 

59019 

40981 

61220 

38780 

63345 

36655 

44 

17 

59056 

401)44 

61256 

38744 

63379 

36621 

43 

18 

59094 

40906 

61292 

38708 

63414 

36586 

42 

19 

59131 

40869 

61328 

38672 

63449 

36551 

41 

20 

9.59168 

10.40882 

9.61864 

10.38636 

9.63484 

10.36516 

40 

21 

59205 

40795 

61400 

38600 

63519 

36481 

39 

22 

59243 

40757 

61436 

38564 

63553 

36447 

38 

23 

59280 

40720 

61472 

38528 

63588 

86412 

37 

24 

59317 

40683 

61508 

3-492 

63623 

36377 

36 

25 

59354 

40646 

61544 

38456 

63657 

36313 

35 

26 

59391 

40609 

61579 

38421 

63692 

36808 

34 

27 

59429 

40571 

61615 

38385 

63726 

36274 

33 

28 

59466 

40534 

61651 

38349 

63761 

36239 

32 

29 

59503 

40497 

61687 

38313 

63796 

36204 

31 

80 

9.59540 

10.40460 

9.61722 

10.38278 

9.63830 

10.36170 

30 

31 

59577 

40423 

61758 

38242 

63865 

36135 

29 

32 

59614 

40386 

61794 

38206 

63899 

36101 

28 

33 

59651 

40349 

61830 

38170 

63934 

86066 

27 

34 

59688 

40312 

61865 

38135 

63968 

36032 

26 

35 

59725 

40275 

61901 

38099- 

64003 

35997 

25 

36 

59762 

40238 

61936 

38064 

64037 

35963 

24 

37 

59799 

40201 

61972 

38028 

64072 

35928 

23 

38 

59835 

40165 

62008 

37992 

64106 

35894 

22 

39 

59872 

40128 

62043 

37957 

64140 

35860 

21 

40 

9.59909 

10.40091 

9.62079 

10  37921 

9.64175 

10.35825 

20 

41 

59946 

40054 

62114 

87886 

64209 

35791 

19 

42 

59983 

40017 

62150 

37850 

64243 

35757 

18 

43 

60019 

39981 

62185 

37815 

64278 

35722 

17 

44 

60056 

39944 

62221 

37779 

64312 

35688 

16 

45 

60093 

39907 

6-2256 

37744 

64346 

35654 

15 

46 

60130 

39870 

62292 

37708 

64381 

35619 

14 

47 

60166 

39834 

62327 

37673 

64415 

35585 

13 

48 

60203 

39797 

62362 

37638 

64449 

35551 

12 

49 

60240 

39760 

62398 

37602 

64483 

35517 

11 

50 

9.60276 

10.39724 

9.62433 

10.37567 

9.64517 

10.35483 

10 

51 

60313 

39687 

62468 

37532 

64552 

35448 

9 

52 

60349 

39651 

62504 

37496 

64586 

35414 

8 

53 

60386 

39614 

62539 

37461 

64620 

35380 

7 

54 

60422 

39578 

62574 

37426 

64654 

35346 

6 

55 

60459 

39541 

62609 

37391 

64688 

35312 

5 

56 

60495 

39505 

62645 

37355 

64722 

35278 

4 

57 

60532 

30468 

62680 

37320 

64756 

35244 

3 

58 

60568 

39432 

62715 

37285 

64790 

35210 

2 

59 

60605 

39395 

62750 

37250 

64824 

35176 

1 

60 

60641 

39359 

62785 

37215 

64858 

35142 

0 

t 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

, 

68« 

67° 

66° 

312 


TABLE  XXI.— LOG.  TANGENTS   AND   COTANGENTS. 


24° 

25° 

26° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.64858 

10.35142 

9.66867 

10.33133 

9.68818 

10.31182 

60 

1 

64892 

35108 

66900 

33100 

68850 

31150 

59 

2 

64926 

35074 

66933 

33067 

68882 

31118 

58 

3 

64960 

35040 

66966 

33034 

68914 

31086 

57 

4 

G4994 

35006 

66999 

33001 

68946 

31054 

56 

5 

K038 

34972 

67032 

32968 

68978 

31022 

85 

6 

65062 

34938 

67065 

32935 

69010 

30990 

54 

7 

65096 

34904 

67098 

32902 

69042 

30958 

53 

8 

65130 

34870 

67131 

32869 

69074 

30926 

52 

9 

65164 

34836 

67163 

32837 

69106 

30894 

61 

10 

9.65197 

10.34803 

9.67196 

10.32804 

9.69138 

10.30862 

50 

11 

65231 

34769 

67229 

32771 

69170 

30830 

49 

12 

65265 

34735 

67262 

32738 

69202 

80798 

48 

13 

65299 

34701 

67295 

32705 

69234 

30766 

47 

14 

65333 

34667 

67327 

32673 

69266 

30734 

46 

15 

65366 

34634 

67360 

32640 

69298 

30702 

45 

16 

65400 

34600 

67393 

32607 

69329 

30671 

44 

17 

65434 

34566 

67426 

82574 

69361 

30639 

43 

18 

65467 

34533 

67458 

32542 

69393 

30607 

42 

19 

65501 

34499 

67491 

32509 

69425 

30575 

41 

20 

9.65535 

10.34465 

9.67524 

10.32476 

9.69457 

10.30543 

40 

21 

65568 

34432 

67556 

32444 

69488 

30512 

39 

22 

65602 

34398 

67589 

32411 

69520 

30480 

38 

23 

65636 

34364 

67622 

32378 

69552 

30448 

37 

24 

656G9 

34331 

67654 

32346 

69584 

30416 

36 

25 

65703 

34297 

67687 

32313 

69615 

30385 

35 

26 

65736 

34264 

67719 

32281 

69647 

30353 

34 

27 

65770 

34230 

67752 

32248 

69679 

30321 

33 

28 

65803 

34197 

67785 

32215 

69710 

30290 

32 

n 

65837 

34163 

67817 

32183 

69742 

30258 

31 

30 

9.65870 

10.34130 

9.67850 

10.32150 

9.69774 

10.30226 

30 

31 

65904 

34096 

67882 

32118 

69805 

30195 

29 

32 

65937 

34063 

67915 

32085 

69837 

30163 

28 

33 

65971 

34029 

67947 

32053 

69868 

30132 

27 

34 

66004 

33996 

67980 

32020 

69900 

30100 

26 

35 

66038 

33962 

68012 

31988 

69932 

30068 

25 

86 

66071 

83929 

68044 

31956 

69963 

30037 

24 

37 

66104 

33896 

68077 

31923 

69995 

30005 

23 

38 

66138 

33862 

68109 

31891 

70026 

29974 

22 

39 

66171 

33829 

68142 

31858 

70058 

29942 

21 

40 

9.66204 

10.33796 

9.68174 

10.31826 

9.70089 

10.29911 

20 

41 

66238 

33702 

68206 

31794 

70121 

29879 

19 

42 

66271 

33729 

68239 

31761 

70152 

29848 

18 

43 

66304 

33696 

68271 

31729 

70184 

29816 

17 

44 

66337 

33663 

68303 

31697 

70215 

29785 

16 

45 

66371 

33629 

68336 

31664 

70247 

29753 

15 

46 

66404 

33596 

68368 

31632 

70278 

29722 

14 

47 

66437 

33563 

68400 

31600 

70309 

29691 

13 

48 

66470 

33530 

68432 

31568 

70341 

29659 

12 

49 

66503 

33497 

68465 

31585 

70372 

29628 

11 

50 

9.66537 

10.33463 

9.68497 

10.31503 

9.70404 

10.29596 

10 

51 

66570 

33430 

68529 

31471 

70435 

29565 

9 

52 

66603 

33397 

68561 

31439 

70466 

29534 

8 

53 

66636 

33364 

68593 

31407 

70498 

29502 

7 

54 

66669 

33331 

68626 

31374 

70529 

29471 

6 

55 

66702 

33298 

68658 

31342 

70560 

29440 

5 

56 

66735 

33265 

68690 

31310 

70592 

29408 

4 

57 

66768 

33232 

68722 

31278 

70623 

29377 

3 

58 

66801 

33199 

68754 

31246 

70654 

29346 

2 

59 

66834 

33166 

68786 

31214 

70685 

29315 

1 

60 

66867 

33133 

68818 

31182 

70717 

29283 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

66° 

64° 

63° 

313 


TABLE  XXI.— LOG.  TANGENTS  AND  COTANGENTS 


/ 

27° 

28° 

29° 

Tan 

Cotan 

Tan 

Cotau 

Tan 

Cotan 

0 

9.70717 

10.29283 

9.72567 

10.27438 

9.74375 

10.25625 

60 

1 

70748 

29252 

72598 

27402 

74405 

26595 

59 

2 

70779 

29221 

72628 

27372 

74435 

255G5  . 

58 

3 

70810 

29190 

72659 

27341 

74465 

25585 

57 

4 

70841 

29159 

72689 

27311 

74494 

25306 

56 

5 

70873 

29127 

72720 

27280 

74524 

25476 

55 

6 

70904 

29096 

72750 

27250 

74554 

25446 

54 

7 

70935 

29065 

72780 

27220 

74583 

25417 

53 

8 

70966 

29034 

72811 

27189 

74613 

25387 

52 

9 

70997 

29003 

72811 

27159 

74643 

25357 

51 

10 

9.71028 

10.28972 

9.72872 

10.27128 

9.74673 

10.25327 

50 

11 

71059 

28941 

72902 

27098 

74702 

25298 

49 

12 

71090 

28910 

72932 

27068 

74732 

252G8 

48 

13 

71121 

28879 

72963 

27037 

74762 

25238 

47 

14 

71153 

28847 

72993 

27007 

74791 

252U9 

46 

15 

71181 

28816 

73023 

26977 

74821 

251  7  9 

45 

16 

71215 

28785 

73054 

26946 

74851 

25149 

44 

17 

71246 

28754 

73084 

26916 

74880 

25120 

43 

18 

71277 

28723 

73114 

26886 

74910 

25090 

42 

19 

71303 

28692 

73144 

26856 

74939 

2506  1 

41 

20 

9.71339 

10.28661 

9.73175 

10.26825 

9.74969 

10.25031 

40 

21 

71370 

28630 

73205 

26795 

74998 

25002 

39 

22 

71401 

28599 

73235 

26765 

75028 

24972 

38 

23 

71431 

28569 

73265 

26735 

75058 

24942 

37 

24 

71462 

28538 

73295 

26705 

75(87 

24913 

36 

25 

71493 

28507 

73326 

26674 

75117 

24883 

35 

26 

71524 

28476 

73316 

26644 

75146 

24854 

34 

27 

71555 

28445 

733S6 

26614 

75176 

24824 

33 

28 

715S6 

28414 

7341  G 

26584 

75205 

24795 

32 

29 

71617 

28383 

73446 

26554 

75235 

24765 

31  • 

30 

9.71648 

10.28352 

9.73476 

10.26524 

9.75264 

10.24736 

30 

31 

71679 

28321 

73507 

26493 

75294 

24706 

29 

32 

71709 

28291 

73537 

26463 

75323 

24677 

28 

33 

71740 

28260 

73567 

26433 

75353 

24647 

27 

34 

71771 

28229 

73597 

26403 

75382 

24618 

26 

35 

71802 

28198 

73627 

26373 

75411 

24589 

25 

36 

71833 

28167 

73G57 

26M3 

75441 

24559 

24 

37 

71863 

28137 

73687 

26313 

75470 

24530 

23 

38 

71894 

2810o 

73717 

26283 

75500 

24500 

39 

71925 

28075 

73747 

26253 

75529 

24471 

21 

40 

9.71955 

10.28045 

9.73777 

10.26223 

9.75558 

10.24442 

20 

41 

71986 

28014 

73807 

26193 

75588 

24412 

19 

42 

72017 

27983 

73837 

26163 

75617 

24383 

18 

43 

72048 

27958 

73867 

26133 

75647 

24353 

17 

44 

72078 

27922 

73897 

26103 

75676 

24824 

16 

45 

72109 

27891 

73927 

26073 

75705 

24295 

15 

46 

72140 

27860 

73957 

26043 

75735 

2421  :5 

14 

47 

72170 

27830 

73987 

26013 

75764 

24236 

13 

48 

72201 

27799 

74017 

259S3 

75793 

24207 

12 

49 

72231 

27769 

74047 

25953 

75822 

24178 

11 

50 

9.72262 

10.27738 

9.74077 

10.25923 

9.7SH52 

10.24148 

10 

51 

72293 

27707 

74107 

25893 

75881 

24119 

9 

52 

72323 

27677 

74137 

25863 

75910 

24080 

8 

53 

72354 

27646 

74166 

25834 

75939 

24061 

54 

723S4 

27616 

7419(5 

25R04 

75969 

24031 

6 

55 

72415 

27585 

74888 

2o774 

75998 

24002 

5 

56 

72445 

27555 

74256 

25744 

76027 

23973 

4 

57 

72476 

27524 

74286 

25714 

76056 

23944 

3 

58 

72506 

27494 

74316 

25684 

76086 

23014 

2 

59 

72537 

27403 

74345 

25655 

76115 

2388") 

1 

60 

72567 

27433 

74375 

25625 

76144 

23856 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

62° 

61° 

60° 

314 


TABLE  XXL— LOG.  TANGENTS   AND  COTANGENTS. 


/ 

30° 

31° 

32° 

f 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

0 

9.76144 

10.23856 

9.77877 

10.22123  ' 

9.79579 

10.20421 

60 

76173 

23827 

77906 

22094 

79607 

20393 

59 

2 

76202 

23798 

77935 

22065 

79635 

20365 

58 

3 

76231 

23769 

77963 

22037 

79663 

20337 

57 

4 

76261 

23739 

77992 

22008 

79691 

20309 

56 

5 

76290 

23710 

78020 

21980 

79719 

20281 

55 

6 

76319 

23681 

78049 

21951 

79747 

20253 

54 

7 

76348 

23652 

78077 

21923 

79776 

20224 

53 

8 

76377 

23623 

78106 

21894 

79804 

20  '96 

52 

9 

76406 

23594 

78135 

21865 

79882 

20168 

51 

10 

9.76435 

10.23565 

9.78163 

10.21837 

9.79860 

10.20140 

50 

11 

76464 

23536 

78192 

21808 

79888 

20112 

49 

12 

76493 

23507 

78220 

21780 

79916 

20084 

48 

13 

76522 

23478 

78249 

21751 

79944 

20056 

47 

14 

76551 

23449 

78277 

21723 

79972 

20028 

46 

15 

76580 

23420 

78306 

21694 

80000 

20000 

45 

16 

76609 

23391 

78334 

21666 

80028 

19972 

44 

17 

76639 

23361 

78363 

21637 

80056 

19944 

43 

18 

76668 

23332 

78391 

21609 

80084 

19916 

42 

19 

76697 

23303 

78419 

21581 

80112 

19888 

41 

20 

9.76725 

10.23275 

9.78448 

10.21552 

9.80140 

10.19860 

40 

21 

76754 

23246 

78476 

21524 

80168 

19832 

39 

22 

76783 

23217 

78505. 

21495 

80195 

19805 

38 

23 

76812 

23188 

78533 

21467 

80223 

19777 

37 

24 

76841 

23159 

78562 

21438 

80251 

19749 

36 

25 

76870 

23180 

78590 

21410 

80279 

19721 

35 

26 

76899 

23101 

78618 

21382 

80307 

19693 

34 

27 

76928 

23072 

78647 

21353 

80335 

19665 

38 

28 

76957 

23043 

78675 

21325 

80363 

19637 

32 

29 

76986 

23014 

78704 

21296 

80391 

19609 

31 

30 

9.77015 

10.22985 

9.78732 

10.212G8 

9.80419 

10.19581 

30 

31 

77044 

22956 

78760 

21240 

80447 

19553 

29 

32 

77073 

22927 

78789 

21211 

80474 

19526 

28 

33 

77101 

22899 

78817 

21183 

80502 

19498 

27 

34 

77130 

22870 

78845 

21155 

80530 

19470 

26 

35 

77159 

22841 

78874 

21126 

80558 

19442 

25 

36 

771  88 

22812 

78902 

21098 

80586 

19414 

24 

37 

77217 

22783 

78930 

21070 

80614 

19386 

23 

38 

77246 

22754 

78959 

21041 

80642 

19358 

22 

39 

77274 

22726 

78987 

21013 

80669 

19331 

21 

40 

9.77303 

10.22697 

9.79015 

10.20985 

9.80697 

10.19303 

20 

41 

77332 

22668 

79043 

20957 

80725 

19275 

19 

42 

77361 

22639 

79072 

20928 

80753 

19247 

18 

43 

77390 

22610 

79100 

20900 

80781 

19219 

17 

44 

77418 

22582 

79128 

20872 

80808 

19192 

16 

45 

77447 

22553 

79156 

20844 

80836 

19164 

15 

46 

77476 

22524 

79185 

20815 

80864 

19136 

14 

47 

77505 

22495 

79213 

20787 

80892 

19108 

13 

48 

77533 

22467 

79241 

20759 

80919 

19081 

12 

49 

77562 

22438 

79269 

20731 

80947 

19053 

11 

50 

9.77591 

10.22409 

9.79297 

10.20703 

9.80975 

10.19025 

10 

51 

77019 

22381 

79326 

20674 

81003 

18997 

9 

52 

77648 

22352 

79354 

20646 

81030 

18970 

8 

53 

77677 

22323 

79382 

20618 

81058 

18942 

7 

54 

77706 

22294 

79410 

20590 

81086 

18914 

6 

55 

77734 

22266 

79438 

20562 

81113 

18887 

5 

56 

77763 

22237 

79466 

20534 

81141 

18859 

4 

57 

77791 

22209 

79495 

20505 

81169 

18831 

3 

58 

77820 

22180 

79523 

20477 

81196 

18804 

2 

59 

77849 

22151 

79551 

20449 

81224 

18770 

1 

60 

77877 

22123 

79579 

20421 

81253 

18748 

0 

, 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

69° 

58° 

67° 

315 


TABLE  XXI.— LOG.  TANGENTS   AND   COTANGENTS. 


33° 

34° 

35° 

• 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.81252 

10.18748 

9.82899 

10.17101 

9.84523 

10.15477 

60 

1 

81279 

18721 

82926 

17074 

81550 

15450 

59 

2 

81307 

18693 

82953 

17047 

8J576 

15424 

58 

g 

81335 

18665 

82980 

17020 

84603 

15397 

57 

4 

81362 

18638 

83008 

16992 

84630 

15370 

56 

g 

81390 

18610 

83035 

16965 

84657 

15343 

55 

81418 

18582 

83062 

16938 

84684 

15316 

54 

7 

81445 

18555 

83089 

16911 

84711 

15289 

53 

g 

81473 

18527 

83117 

16883 

84738 

152G2 

52 

9 

81500 

18500 

83144 

16856 

84764 

15236 

51 

10 

9.81528 

10.18472 

9.83171 

10.16829 

9.84791 

10.15209 

50 

81556 

18444 

83198 

16802 

84818 

15182 

49 

12 

81583 

18417 

83225 

16775 

84845 

15155 

48 

18 

81611 

18389 

83252 

16748 

84872 

15128 

47 

14 

81638 

18362 

83280 

16720 

84899 

15101 

46 

•JK 

81666 

18334 

83307 

16693 

84925 

15075 

45 

JO 

81693 

18307 

83334 

16666 

84952 

15048 

44 

17 

81721 

18279 

83361 

16639 

84979 

15021 

43 

10 

81748 

18252 

83338 

16612 

85006 

14994 

42 

JO 

19 

81776 

18224 

83415 

16585 

85033 

14967 

41 

20 

9.81803 

10.18197 

9.83142 

10.16558 

9.85059 

10.14941 

40 

21 

81831 

18169 

83470 

16530 

85086 

14914 

39 

81858 

18142 

83497 

.  16503 

85113 

14887 

38 

23 

81886 

18114 

83524 

16476 

85140 

14860 

37 

24 

81913 

18087 

83551 

16449 

85166 

14834 

36 

25 

81941 

18059 

83578 

16422 

85193 

14807 

35 

26 

81968 

18032 

83605 

16395 

85220 

14780 

34 

27 

81996 

18004 

83632 

16368 

85247 

14753 

33 

2H 

82023 

17977 

83659 

16341 

85273 

14727 

32 

*o 

29 

82051 

17949 

83686 

16314 

85300 

14700 

31 

30 

9.82078 

10.17922 

9.83713 

10.16287 

9.85327 

10.14673 

30 

31 

82106 

17894 

83740 

16260 

85354 

14646 

29 

0.1 

82133 

17867 

83768 

16232 

85380 

14620 

28 

o-* 

83 

82161 

17839 

83795 

16-205 

85407 

14593 

27 

82188 

17812 

83822 

16178 

85434 

14566 

26 

OK 

82215 

17785 

83849 

16151 

85460 

14540 

25 

OD 

36 

82243 

17757 

83876 

16124 

85487 

14513 

24 

07 

82270 

17730 

83903 

16097 

85514 

14486 

23 

ul 

00 

82298 

17702 

83930 

16070 

85540 

14460 

22 

OO 

39 

82325 

17675 

83957 

16043 

85567 

14433 

21 

40 

9.82352 

10.17648 

9.83984 

10.16016 

9.85594 

10.14406 

20 

« 

82380 

17620 

84011 

15989 

85620 

14380 

19 

42 

82407 

17593 

84038 

15962 

85647 

14353 

18 

82435 

17565 

84065 

15935 

85674 

14326 

17 

AA 

82462 

17538 

84092 

15908 

85700 

14300 

16 

44 

82489 

17511 

84119 

15881 

85727 

14273 

15 

4fi 

82517 

17483 

84146 

15854 

85754 

14246 

14 

40 

47 

82544 

17456 

84173 

15827 

85780 

14220 

13 

48 

82571 

17429 

84200 

15800 

85807 

14193 

12 

4o 

49 

82599 

17401 

84227 

15773 

85834 

14166 

11 

^ 

9.82626 

10.17374 

9.84254 

10.15746 

9.85860 

10.14140 

10 

82653 

17347 

84280 

15720 

85887 

14113 

9 

82681 

17319 

84307 

15693 

85913 

14087 

8 

fcO 

82708 

17292 

84334 

15666 

85940 

14060 

7 

Oo 
fr  1 

82735 

17265 

84361 

15639 

85967 

14033 

6 

04 

55 

82762 

17238 

843S8 

15612 

85993 

14007 

5 

KC 

82790 

17210 

84415 

15585 

86020 

13980 

4 

00 

82817 

17183 

84442 

15558 

86046 

13954 

3 

KQ 

82844 

17156 

84469 

15531 

86073 

13927 

2 

Oo 

en 

82871 

17129 

84496 

15504 

86100 

13900 

1 

oy 
60 

82899 

17101 

84523 

15477 

86126 

13874 

0 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

, 

50° 

55° 

54° 

316 


TABLE  XXI.— LOG.  TANGENTS  AND   COTANGENTS. 


86° 

, 

J7° 

38° 

' 

Tan    Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.86126  10. 

13874 

9.87711 

10.12289 

9.89281 

10.10719 

60 

1 

86153 

13847 

87738 

12262 

89307 

10693 

59 

2 

86179 

13821 

87764 

12236 

89333 

10667 

58 

3 

86206 

13794 

87790 

12210 

89359 

10641 

57 

4 

86232 

13768 

87817 

12183 

89385 

10615 

56 

5 

86259 

13741 

87843 

12157 

89411 

10589 

55 

6 

86285 

13715 

87869 

12131 

89437 

10563 

54 

7 

86312 

13688 

87895 

12105 

89463 

10537 

53 

8 

86333 

13662 

87922 

12078 

89489 

10511 

52 

9 

86365 

13635 

87948 

12052 

89515 

10485 

51 

10 

9.86392  10. 

13603 

9.87974 

10.12026 

9.89541 

10.10459 

50 

11 

86418 

13582 

88000 

12000 

89567 

10433    49 

12 

86445 

13555 

88027 

11973 

89593 

10407 

48 

13 

86471 

13529 

88053 

11947 

89619 

10381 

47 

14 

86498 

13502 

88079 

11921 

89645 

10355 

46 

15 

86524 

13476 

88105 

11895 

89671 

10329 

45 

16 

86551 

13449 

88131 

11869 

89697 

10303 

44 

17 

86577 

13423 

88158 

11842 

89723 

10277 

43 

18 

86603 

13397 

88184 

11816 

89749 

10251 

42 

19 

86630 

13370 

88210 

11790 

89775 

10225 

41 

20 

9.86656  10. 

13344 

9.88236 

10.11764 

9.89801 

10.10199 

40 

21 

86683 

13317 

88262 

11738 

89827 

10173 

39 

22 

86709 

13291 

88289 

11711 

89853 

10147 

38 

23 

86736 

13264 

88315 

11685 

89879 

10121 

37 

24 

86762 

13238 

88341 

11659 

89905 

10095 

36 

25 

86789 

13211 

8S367 

11633 

89931 

10069 

35 

26 

86815 

13185 

8S393 

11607 

89957 

10043 

34 

27 

86842 

13158 

88420 

11580 

89983 

10017 

33 

28 

86868 

13132 

88446 

11554 

90009 

09991 

32 

29 

86894 

13106 

88472 

11528 

90035 

09965 

31 

30 

9.80921  10. 

13079 

9.88498 

10.11502 

9.90081 

10.09939 

30 

31 

86947 

130;>3 

88524 

11476 

90086 

09914 

29 

32 

86974 

13026 

88550 

11450 

90112 

09888 

28 

33 

87000 

13000 

88577 

11423 

90138 

09862 

27 

34 

87027 

12973 

88603 

11307 

90164 

09836 

26 

35 

87053 

12947 

88629 

11371 

90190 

09810 

25 

36 

87079 

12921 

88655 

11345 

90216 

09784 

24 

37 

87106 

12894 

88681 

11319 

90242 

09758 

23 

38 

87132 

12868 

88707 

11293 

90268 

09732 

22 

39 

87158 

12842 

88733 

11267 

90294 

09706 

21 

40 

9.87185  10. 

12815 

9.88759 

10.11241 

9.90320 

10.09680 

20 

41 

87211 

12789 

88786 

11214 

90346 

09654 

19 

42 

87238 

12762 

88812 

11188 

90371 

09629 

18 

43 

87264 

12736 

88838 

11162 

90397 

09603 

17 

44 

45 

87290 
87317 

12710 
12683 

88864 
88890 

11136 
11110 

90423 
90449 

09577 
09551 

16 
15 

46 

87343 

12657 

88916 

11084 

90475 

09525 

14 

47 

87369 

12631 

88942 

11058 

•  90501 

09499 

13 

48 

87396 

12604 

88968 

11032 

90527 

09473 

12 

49 

87422 

12578 

88994 

11006 

90553 

09447 

11 

50 

9.87448  10. 

12552 

9.89020 

10.10980 

9.90578 

10.09422 

10 

51 

87475 

12525 

89046 

10954 

90604 

09396 

9 

52 

87501 

12499 

89073 

10927 

90680 

09370 

8 

53 

87527 

12473 

89099 

10901 

90656 

09344 

7 

54 

87554 

12446 

89125 

10875 

90682 

09318 

6 

55 

87580 

12420 

89151 

10849 

90708 

09292 

5 

56 

87606 

12394 

89177 

10823 

90734 

09266 

4 

57 

87633 

12367 

89203 

10797 

90759 

09241 

3 

58 

87659 

12341 

89229 

10771 

90785 

09215 

2 

59 

87685 

12315 

89255 

10745 

90811 

09189 

1 

60 

87711 

12289 

89281 

10719 

90837 

09163 

0 

,     Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

t 

53° 

62° 

51° 

317 


TABLE  XXI.— LOG.  TANGENTS  AND  COTANGENTS- 


t 

39° 

40° 

41° 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.90837 

10.09163 

9.92381 

10.07619 

9.93916 

10.06084 

60 

1 

90863 

09137 

92407 

07593 

93942 

06058 

59 

2 

90889 

09111 

92433 

07567 

93967 

06033 

58 

3 

90914 

09086 

92458 

07542 

93993 

06007 

57 

4 

90940 

09060 

92484 

07516 

94018 

0598-2 

56 

5 

90966 

09034 

92510 

07490 

94044 

05956 

55 

6 

90992 

09008 

92535 

07465 

94069 

05931 

54 

7 

91018 

08982 

92561 

07439 

94095 

05905 

53 

8 

91043 

08957 

92587 

07413 

94120 

05880 

52 

9 

91069 

08931 

92612 

07388 

94146 

05854 

51 

10 

9.91095 

10.08905 

9.92638 

10.07362 

9.94171 

10.058-29 

50 

11 

91121 

08879 

92663 

07337 

94197 

05803 

49 

12 

91147 

08853 

92689 

07311 

94222 

05778 

48 

13 

91172 

088-28 

92715 

07285 

94248 

05752 

47 

14 

91198 

08802 

92740 

07260 

94273 

05727 

46 

15 

91224 

08776 

92766 

07234 

94299 

05701 

45 

16 

91250 

08750 

92792 

07208 

94324 

05676 

44 

17 

91276 

08724 

92817 

07183 

94350 

05650 

43 

18 

91301 

08699 

92843 

07157 

94375 

056-25 

42 

19 

91327. 

08673 

92868 

07132 

94401 

05599 

41 

20 

9.91353 

10.08647 

9.92894 

10.07106 

9.94426 

10.05574 

40 

21 

91379 

08621 

92920 

07080 

94452 

05548 

39 

22 

91404 

08596 

92945 

07055 

94477 

05523 

38 

23 

91430 

08570 

92971 

07029 

94503 

05497 

37 

24 

91456 

08544 

92996 

07004 

94528 

05472 

36 

25 

91482 

08518 

93022 

06978 

94554 

05446 

35 

26 

91507 

08493 

93048 

06952 

94579 

054-21 

34 

27 

91533 

08467 

93073 

06927 

94604 

05396 

33 

28 

91559 

08441 

93099 

06901 

94630 

05370 

32 

29 

91585 

08415 

93124 

06876 

94655 

05345 

31 

30 

9.91610 

10.08390 

9.93150 

10.06850 

9.94681 

10.05319 

30 

31 

91636 

08364 

93175 

06825 

94706 

05294 

29 

32 

91662 

08338 

93201 

06799 

94732 

05268 

28 

33 

91688 

08312 

93227 

06773 

94757 

05243 

27 

34 

91713 

08-287 

93252 

06748 

94783 

05217 

26 

35 

91739 

08261 

93278 

06722 

94808 

05192 

25 

36 

91765 

08235 

93303 

06697 

94834 

05166 

24 

87 

91791 

08209 

93329 

06671 

94859 

05141 

23 

38 

91816 

08184 

93354 

06646 

94884 

05116 

22 

39 

91842 

08158 

93380 

06620 

94910 

05090 

21 

40 

9.91868 

10.08132 

9.93406 

10.06594 

9.94935 

10.05065 

20 

41 

91893 

08107 

93431 

06569 

94961 

05039 

19 

42 

91919 

08081 

93457 

06543 

94986 

05014 

18 

43 

91945 

08055 

93482 

06518 

95012 

04988 

17 

44 

91971 

08029 

93508 

06492 

95037 

04963 

16 

45 

91996 

08004 

93533 

06467 

95062 

04936 

15 

46 

92022 

07978 

93559 

06441 

95088 

04912 

14 

47 

92048 

07952 

93584 

06416 

95113 

04887 

13 

48 

92073 

07927 

93610 

06390 

95139 

04861 

12 

49 

92099 

07901 

93636 

06361 

95164 

04836 

11 

50 

9.92125 

10.07875 

9.93661 

10.06339 

9.95190 

10.04810 

10 

51 

92150 

07850 

93687 

06313 

95215 

04785 

9 

52 

92176 

07824 

93712 

06288 

95240 

04760 

8 

53 

92202 

07798 

93738 

06262 

95266 

04734 

7 

54 

92227 

07773 

93763 

06237 

95291 

04709 

6 

55 

92253 

07747 

93789 

06211 

95317 

04683 

5 

56 

92279 

07721 

93814 

06186 

95342 

04658 

4 

57 

9-2304 

07696 

93840 

06160 

95368 

04032 

3 

58 

9-2330 

07670 

93865 

06135 

95393 

04607 

2 

59 

92356 

07644 

93891 

06109 

95418 

0458-2 

1 

60 

92381 

07619 

93916 

06084 

95444 

04556 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

., 

50- 

49° 

48° 

318 


FABLE  XXI.— LOG.  TANGENTS   AND   COTANGENTS. 


42° 

43° 

440 

/ 

Tan 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

o 

9.95444 

10.04556 

9.96966 

10.03034 

9.98484 

10.01516 

60 

1 

95469 

04531 

96991 

03009 

98509 

01491 

&9 

3 

95495 

04505 

97016 

02984 

98534 

01466 

58 

3 

95520 

04480 

97042 

02958 

98560 

01440 

67 

4 

95545 

04455 

97067 

02933 

98585 

01415 

56 

5 

95571 

04429 

97092 

02908 

98610 

01390 

55 

6 

95596 

04404 

97118 

02882 

98635 

01365 

54 

7 

95622 

04378 

97143 

02857 

98661 

01339 

53 

8 

95647 

04353 

97168 

02832 

98686 

01314 

52 

9 

95672 

04328 

97193 

02807 

98711 

01289 

51 

10 

9.95698 

10.04302 

9.97219 

10.02781 

9.98737 

10.01263 

50 

i  11 

95723 

04277 

97244 

02756 

98762 

01238 

49 

12 

95748 

04252 

97269 

02731 

98787 

01213 

48 

13 

95774 

04226 

97295 

02705 

98812 

01188 

47 

14 

95799 

04201 

97320 

02680 

98838 

01162 

46 

15 

95825 

04175 

97345 

02655 

98863 

01137 

45 

16 

95850 

04150 

97371 

02629 

98888 

01112 

44 

17 

95875 

04125 

97396 

02604 

98913 

01087 

43 

18 

95901 

04099 

97421 

02579 

98939 

010G1 

42 

19 

95926 

04074 

97447 

02553 

98964 

01036 

41 

20 

9.95952 

10.04048 

9.97472 

10.02528 

9.98989 

10.01011 

40 

21 

95977 

04023 

97497 

02503 

99015 

00985 

39 

22 

96002 

03998 

97523 

02477 

99040 

00960 

38 

23 

96028 

03972 

97548 

02452 

99065 

00935 

37 

24 

96068 

03947 

97573 

0242? 

99090 

00910 

86 

25 

96078 

03922 

97598 

02402 

99116 

00884 

35 

26 

96104 

03896 

97624 

02376 

99141 

00859 

34 

27 

96129 

03871 

97649 

02351 

99166 

00834 

33 

28 

96155 

03845 

97674 

02326 

99191 

00809 

32 

29 

96180 

03820 

97700 

02300 

99217 

00783 

31 

30 

9.96205 

10.03795 

9.97725 

10.02275 

9.99242 

10.00758 

30 

31 

96281 

03769 

97750 

02250 

99267 

00733 

29 

32 

96256 

03744 

97776 

02224 

99293 

00707 

28 

33 

9(5281 

03719 

97801 

02199 

99318 

00682 

27 

34 

96307 

03693 

97826 

02174 

99343 

00657 

26 

as 

96332 

03668 

97851 

02149 

99368 

00632 

25 

36 

96357 

03643 

97877 

02123 

99394 

00606 

24 

37 

96383 

03617 

97902 

02098 

99419 

OC581 

23 

38 

96408 

03592 

97927 

02073 

99444 

00556 

22 

39 

96433 

03567 

97953 

02047 

99469 

00531 

21 

40 

9.96459 

10.03541 

9.97978 

10.02022 

9.99495 

10.00505 

20 

41 

96484 

03516 

9b003 

01997 

99520 

00480 

19 

42 

96510 

03490 

98029 

01971 

99545 

00455 

18 

43 

96535 

03465 

98054 

01946 

99570 

00430 

17 

44 

96560 

03440 

98079 

01921 

99596 

00404 

16 

45 

96586 

03414 

98104 

01896 

99621 

00379 

15 

46 

96611 

03389 

98130 

01870 

99646 

00354 

14 

47 

96636 

03364 

98155 

01845 

99672 

00328 

13 

48 

96662 

03338 

98180 

01820 

99697 

00303 

12 

49 

96687 

03313 

98206 

01794 

99722 

00278 

11 

50 

9.96712 

10.03288 

9.98231 

10.01769 

9.99747 

10.00253    10 

51 

96738 

03262 

98256 

01744 

99773 

00227  ;   9 

52 

96763 

03237 

98281 

01719 

99798 

00202    8 

53 

96788 

03212 

98307 

01693 

99823 

00177    7 

54 

96814 

03186 

98332 

01668 

99848 

00152 

6 

65 

96839 

03161 

98357 

01643 

99874 

00126 

5 

56 

96864 

03136 

98383 

01617 

99899 

00101 

4 

57 

96890 

03110 

98408 

01592 

99924 

00076 

3 

58 

96915 

03085 

98433 

01567 

89949 

00051 

2 

59 

96940 

03060 

98458 

01542 

99975 

00025 

1 

60 

96966 

03034 

98484 

01516 

10.00000 

00000 

0 

/ 

Cotan 

Tan 

Cotan 

Tan 

Cotan 

Tan 

/ 

I 

47° 

46° 

45° 

319 


or  THE 
UNIVERSITY 

OF 


APPENDIX. 


THE  TRANSITION  CURVE. 

The  true  transition  curve  is  a  spiral  whose  radius  of  curva- 
ture at  the  origin,  A  ,  is  infinity,  and  at  any  point  on  the  curve 
is  inversely  proportional  to  the  distance  of  that  point  from  A  ; 
or,  the  degree  of  curvature  is  directly  proportional  to  this  dis- 
tance. The  degree  of  curvature  will  therefore  have  a  constant 
increase  per  foot. 

Let  n  =  this  increase  in  minutes, 

d  =  the  distance  of  any  point  from  A, 
and  8  =  degree  of  curvature  at  distance'  d  ; 

then  8  =  j^,  (1) 

and  d  =  ™*.  (2) 

n 

Now,  since  the  curvature  increases  uniformly  from  0  to  8  in 
the  distance  d,  it  is  evident  that  the  total  angle  turned  will  be 
only  half  that  turned  by  a  8°  curve  d  feet  in  length.  Therefore 
the  total  angle  turned  by  the  spiral  in  the  distance  d  will  be 

60  Bd  , 

(expressed  in  minutes). 
200 

If  in  Fig.  1  B  be  the  point  where  the  degree  of  curvature 
becomes  8,  then 

AB  =  d, 


Substituting  the  value  of  8  from  (1),  we  have 

=^.  (4) 

200 

321 


322 


APPENDIX. 


E 


This  spiral  possesses  the  following  well-known  properties  :  — 

1.  It  is  almost  identical  with  the  cubic  parabola,  the  only 
difference  being  this :  in  the  cubic  parabola  the  ordinates  vary 
as  the  cubes  of  the  abscissas,  while  in  this  curve  they  vary  as 
the  cubes  of  the  corresponding  lengths  of  the  curve. 

2.  The  spiral  bisects  the  offset,  FN,  to  the  central  curve 
produced  and  is  bisected  by  it. 

3.  It  therefore  follows  that  the  central  ordinate  FB  =  £  and 
the  central  curve  offset  FN  =  \  of  the  terminal  ordinate  CE. 

Now,  in  the  cubic  parabola,  the  tangent  of  any  deflection 
angle,  as  BAF,  is  equal  to  £  of  the  tangent  of  the  tangential 
angle  BGF.  And  since  small  angles  vary  nearly  as  their  tan- 
gents, we  may  assume  that 

(5) 


and  since 


we  have 


600 
=  BGF-BAF, 

=  ^L_^L  =  ^. 
200       600       300* 


(6) 


*  Fig.  2  exhibits  lines  in  their  proper  relation.    Fig.  1  is  not  drawn 
to  scale  but  made  to  aid  in  the  demonstration. 


APPENDIX.  323 

By  (5)  and  (6)  the  spiral  may  be  laid  out  with  the  transit 
using  (5)  for  deflections  from  the  tangent  and  (6)  for  turn- 
ing tangent  from  any  chord.  If  there  be  no  obstacle  in  the 
line  of  sight,  and  the  terminating  spiral  be  run .  backward, 
we  need  nothing  more.  But  it  may  be  necessary  to  set  some 
intermediate  point,  as  B,  and  continue  from  that  point  by 
deflections  from  the  tangent  BK ;  and  it  is  generally  desirable 
to  run  the  terminating  spiral  with  the  transit  set  at  its  junction 
with  the  central  curve.  It  is  therefore  necessary  to  find  a 
general  expression  for  the  deflection  from  any  point  on  the  spiral 
to  any  other  point  on  the  spiral. 

From  (5)  we  have 

.00029  nd2 

smSAF-  *%?"•  <7> 

since  .00029  =  sine  of  1'. 

Then  if  d'  =  BC,  that  is  any  distance  beyond  B,  we  have  also 

*?.  (8) 


Hence  we  have  for  the  ordinates  FB  and  CE,  by  regarding  the 
curves  AB  and  AC  as  equal  to  the  chords  of  the  same, 

.00029  nd* 
ra  =  — 600— '  (9) 


CE  =  -  v    n      ;  .  (10) 

600 


TVion        r'T-T"         C'TT'          7772  v  '  ~L_!~     /  /1 1  \ 

^600~  "'  (U) 

and       ^  -  sin  CBH  -  -00029  n  (3  c?2  +  3  dd'  +  ^/2)  a  . 
d'    ~                                              600 


« 


But        CBK  =  CBH  -  KBH  =  CBH  -  BGF  =  CBH  -  —  . 

200 

Therefore         CBg  =  »<81"'  +  "*>  =  ?g  -H^.  (14, 

BOO  600          200 


324  APPENDIX. 

To  obtain  an  expression  for  the  angle  BCK,  we  have  from  (4) 


and  since  BCK  =  CDS  -  CBH, 


. 


600  300          200 

Eqs.  (14)  and  (16)  may  be  put  into  the  following  forms :  — 
_  /nd    .  1    .  60  d 


\6Q      2       100  /      600 

~  (  ian  X   o    X     inn    /  ~~  ~at\n '  ( -     ) 


in  which  the  first  term  of  (17)  is  the  deflection  for  a  distance 
d'  of  a  circular  curve,  whose  curvature  is  equal  to  the  curvature 
of  the  spiral  at  distance  d  (see  Eq.  1)  ;  and  the  first  term  of 
(18)  is  the  deflection  for  a  distance  d'  of  a  circular  curve,  whose 
curvature  is  equal  to  the  curvature  of  the  spiral  at  distance 
d  -f  d'.  Hence  the  following  property  :  — 

The  deflection  from  tangent  at  any  point  on  the  spiral  to  any 
other  point  on  the  spiral  is  equal  to  the  deflection  of  a  circular  curve 
for  the  same  distance,  whose  curvature  is  equal  to  the  curvature  of 
the  spiral  at  said  tangent  point,  PLUS  or  MINUS  the  deflection  for 
an  equal  distance  from  the  initial  point  of  the  spiral,  according  as 
the  transit  is  turned  TOWARD  or  FROM  the  central  curve. 

To  find  the  length  of  semi-tangent,  T,  and  external  secant,  E, 
we  have  from  (9),  since 

FN  =  2  FB, 

™,0  =  ™, 

in  which  d  =  £  total  length  of  spiral.  Therefore  if  in  Fig.  2 
L  =  total  length  of  spiral,  V—  point  of  intersection  of  tangents, 
P  —  middle  point  of  circular  curve,  and  /  =  angle  of  intersec- 
tion of  tangents,  then 

AV=  T=(R+  0)taniI  +  iL,  (20) 


and  py  =  E=  -R.  (21) 

COS-J  I 


APPENDIX. 


325 


FIG.  2. 


326  APPENDIX. 

It  is  proposed  to  call  n  in  these  formulas  the  number  of  the 
spiral.  Thus  if  n  —  2,  we  would  designate  the  spiral  as  a 
number  two  spiral ;  if  n  =  3,  a  number  three  spiral,  etc.  n  may 
have  any  value  whatever,  either  entire  or  fractional.  In  prac- 
tice its  value  will  generally  be  between  1  and  6. 

It  will  be  seen  that  by  this  method  the  transition  curve 
becomes  absolutely  universal.  By  varying  the  value  of  n  a 
spiral  of  any  length  whatever  can  be  fitted  to  a  central  curve 
of  any  degree.  The  question  of  standard  sub-chords,  which  are 
used  in  a  great  many  systems  of  treating  the  transition  curve, 
is  entirely  eliminated.  A  transit  hub  can  be  set  at  any  point 
on  the  curve  and  deflections  may  be  turned  from  any  point  on 
the  curve  to  any  other  point  with  practically  the  same  ease  and 
facility  as  in  the  simple  curve. 


HOW   TO   LAY   OUT   A   SPIRAL   CURVE. 

1.  The  tables  which  follow  will  greatly  facilitate  the  work  of 
computing  and  laying  out  the  spiral.    Table  I  gives  the  length 
(Z),  total  angle  (^4),  and  central  curve  offset  (0)  of  a  No.  1 
spiral  for  central  curves  of  different  degrees.    To  find  the  cor- 
responding elements  of  a  spiral  of  different  number  it  will  be 
observed  that  L  and  A  vary  inversely  as  the  number  and  0 
inversely  as  the  square  of  the  number.     The  number  of  the 
spiral  should  be  so  chosen  that  the  total  angle  will  not  much 
exceed  15°,  as  the  formula  for  deflections  becomes  less  accu- 
rate for  large  angles.     Table  II  is  a  tabulation  of  Eq.  5  for  a 
No.   1  spiral. 

2.  Determine  by  inspection  of  Table  I  and  the  conditions 
on  the  ground  the  number  of  the  spiral  to  be  used  in  any  par- 
ticular case,  and  find  central  curve  offset,  O,  for  this  spiral. 

Calculate  semi-tangent  by  the  following  formula, 

T  -  (R  +  O)  tan  £  I  +  $  L, 

and  locate  points  of  the  spiral  A  and  F  on  each  tangent. 

3.  Set  transit  at  A  and  by  use  of  Table  II  turn  deflections 
for  all  points  from  A  to  C.     Move  transit  to  C,  backsight  on 


APPENDIX.  327 

A,  and  turn  twice  the  angle  that  was  turned  at  A,  which  will 
bring  transit  on  tangent  at  this  point.  Continue  around  the 
central  curve  to  D  in  the  usual  manner.  The  transit  may  then 
be  moved  to  F  and  the  terminating  spiral  run  backward  in 
the  same  manner  as  the  first  spiral  was  run  forward. 


FIG.  3. 

4.  If  desired  to  run  both  spirals  and  central  curve  continu- 
ously from  beginning  to  end,  or  to  turn  deflections  from  inter- 
mediate points  on  either  spiral,  observe  the  property  of  the. 
spiral,  stated  on  page  324. 

5.  An  example  will  illustrate  the  application  of  this  prop- 
erty.    Suppose  it  be  required  to  run  a  4°  curve  with  Xo.  1 
spirals,  and  the  initial  point,  A ,  be  established  at  Sta.  122  4-  80 
as  shown  in  Fig.  3.     Table  I  shows  that  either  spiral  will  be 
240  feet  long,  and  that  the  initial  spiral  will  connect  with  the 
central  curve  at  Sta.  125  +  20.     Suppose  the  other  end  of  the 
central  curve  be  found  at  Sta.   131  +  60.     The  terminating 
spiral  will  then  unite  with  the  tangent  at  Sta.  134. 

6.  Suppose  it  be  desired  to  set  transit  points  at  B,  Sta.  124, 
and  E,  Sta.  133,  in  addition  to  points  C  and  D  at  ends  of  the 


328  APPENDIX. 

central  curve,   and  run  the   entire  curve   continuously.     Set 
transit  at  A,  and  turn  deflections  as  follows  :  — 

Sta.  123 :  deflection  for  20  feet  (by  Table  II)          0°  00. 7' 
Sta.  124  :  deflection  for  120  feet  (by  table)  0°  24.' 

7.  Move  transit  to  B.    Backsight  on  A  and  turn  0°  48'  for 
tangent.     Then  for  deflections  from  B  to  C,  since  the  spiral  has 
attained  a  curvature  of  2°  per  100  feet  at  B,  we  have,  according 
to  the  property  stated  above, 

Sta.  125 :  deflection  for  100  feet,  2°  curve  1°  00. ' 

Plus  deflection  for  100  feet  spiral  (by  table)  16.7' 

Total ;     1°  16.7' 

Sta.  125  +  20 :  deflection  for  120  feet,  2°  curve       1°  12.' 

Plus  deflection  for  120  feet  spiral  (by  table)  24/ 

Total 1°  36.' 

8 .  Then  move  transit  to  C.     Backsight  on  B,  and  to  turn 
tangent,  since  curvature  has  become  4°  at  C, 

Deflection  for  120  feet,  4°  curve  2°  24.' 

Minus  deflection  for  120  feet  spiral  (by  table)  24.' 

Total 2°  00.' 

9.  Circular  curve  is  then  run  to  D.     Next  move  transit  to 
Z>  and  set  on  tangent  at  this  point.     We  then  have  for  deflec- 
tions from  D  to  F,  since  curvature  is  here  4°, 

Sta.  132  :  deflection  for  40  feet,  4°  curve  0°  48.' 

Minus  deflection  for  40  feet  spiral  (by  table)  02.7' 

Total 0°  45.3' 

Sta.  133  :  deflection  for  140  feet,  4°  curve  2°  48.' 

Minus  deflection  for  140  feet  spiral  (by  table)  32.7' 

Total 2°  15.3' 

10.  Move  transit  to  E.  Backsight  on  D,  and  to  turn  tan- 
gent, since  spiral  has  a  curvature  of  1°  40'  at  this  point,  we 
have 

Deflection  for  140  feet,  1°  40'  curve  1°  10.' 

Plus  deflection  for  140  feet  spiral  (by  table)  32.7' 

Total  .  1°42.7' 


APPENDIX.  -  329 


Then,  to  finish  curve,  we  have 

Deflection  for  100  feet,  1°  40'  curve  0°  50. ' 
Minus  deflection  for  100  feet  spiral  (by  table)  16.  V 

Total 0°33.3/ 

11.    By  moving  transit  to  F,  and  backsighting  upon  E,  we 
have  for  turning  on  final  tangent 

Deflection  for  100  feet  spiral  (by  table)  0°  16.7' 


EXAMPLES. 

1.  Intersection  of  two  tangents,  Sta.  127  +  65.2 ;  inter- 
section angle  /  =  35°  24'. 

Connect  tangents  with  a  6°  curve  (JR  =  955)  terminating  in 
a  No.  2  spiral.  By  reference  to  Table  I  we  find 

L  =  180  feet ;  A  =  5°  24' ;  0  =  1.41  feet. 

.-.  T  =  (B  +  0)  tan  i  I  +  i  L  =  395.2  feet. 
Beginning  of  spiral,  BS  =  Sta.  123  +  70. 

Beginning  of  central  curve,  BC  =    "     125  +  50. 
End  of  central  curve,  EC  =    "     129  +  60. 

End  of  spiral,  ES  =    "     131+  40. 

Set  transit  at  BS,  Sta.  123  +  70 
Deflection  to  Sta.  124  =  0°  03. ' 

Deflection  to  Sta.  125  =  0°  56.4' 

Deflection  to  Sta.  125  +  50          =1°  48. ' 

Move  transit  to  Sta.  125  +  50.  Backsight  on  Sta.  123  +  70 
and  turn  3°  36'  for  tangent.  Run  central  curve  to  Sta.  129  +  60 
in  usual  manner.  Move  transit  to  this  point  and  set  on  tan- 
gent in  usual  way.  Then 

Deflection  to  Sta.  130  =  1°  06.6' 

Deflection  to  Sta.  131  =  3°  06.6' 

Deflection  to  Sta.  131  +40          -  3°  36. ' 

Move  transit  to  Sta.  131  +  40.  Backsight  on  Sta.  129  +  60 
and  turn  on  final  tangent  by  turning  1°  48'. 


330  -   APPENDIX. 

2.  Illustrating  the  use  of  the  spiral  as  a  transition  between 
the  branches  of  a  compound  curve.  A  2°  curve  compounding 
to  a  6°  is  to  be  connected  by  a  No.  1  spiral. 


In  this  case  the  length  of  the  spiral  and  offset  between  curves 
produced  is  the  same  as  for  a  4°  (6°  —  2°)  curve.  L  =  240  feet, 
O  =  l. 67  feet. 

Deflections  from  Sta.  136  +  40  to  Sta.  138  +  80  would  be 
deflections  for  a  2°  curve  plus  tabular  deflections  for  spiral, 
and  deflections  from  Sta.  138  +  80  to  Sta.  136  +  40  would  be 
deflections  for  a  6°  curve  minus  tabular  deflections  for  spiral,  as 
follows :  — 

Transit  at  Sta.  136  +  40  : 

Deflection  to  Sta.  137  =  0°  42. ' 

Deflection  to  Sta.  138  =  2°  18. 1' 

Deflection  to  Sta.  138  +  80  =  4°  OO/ 

Transit  at  Sta.  138  +  80: 

Deflection  to  Sta.  138  =  2°  13.3' 

Deflection  to  Sta.  137  =  4°  30. ' 

Deflection  to  Sta.  136  +  40  =  5°  36. ' 


APPENDIX. 


TABLE  I. 

ELEMENTS  OF  A  No.  1  SPIKAL. 

L  =  Length  of  spiral. 

A  =  Total  angle  turned  by  spiral. 

O  =  Central  curve  offset. 

L  and  A  vary  inversely  as  the  number  of  spiral. 

O  varies  inversely  as  square  of  the  number. 


Deg. 

L 

A 

O 

Deg. 

L 

A 

0 

2°  00' 

120 

1°  12'  00" 

.21 

7°  10' 

430 

15°  24'  30" 

9.61 

10 

130 

1  24  30 

.27 

20 

440 

16  08  00 

10.29 

20 

140 

1  38  00 

.33 

30 

450 

16  52  30 

11.01 

30 

150 

1  52  30 

.41 

40 

460 

17  38  00 

11.76 

40 

160 

2  08  00 

.49 

50 

470 

18  24  30 

12.55 

50 

170 

2  24  30 

.59 

8  00 

480 

19  12  00 

13.36 

3  00 

180 

2  42  00 

.70 

10 

490 

20  00  30 

14.22 

10 

190 

3  00  30 

.83 

20 

500 

20  50  00 

15.10 

20 

200 

3  20  00 

.97 

30 

510 

21  40  30 

16.03 

30 

210 

3  40  30 

1.12 

40 

520 

22  32  00' 

16.99 

40 

220 

4  02  00 

1.29 

50 

530 

23  24  30 

17.99 

50 

230 

4  24  30 

1.47 

9  00 

540 

24  18  00 

19.03 

4  00 

240 

4  48  00 

1.67 

10 

550 

25  12  30 

20.10 

10 

250 

5  12  30 

1.89 

20 

560 

26  08  00 

21.22 

20 

260 

5  38  00 

2.12 

30 

570 

27  04  30 

22.38 

30 

270 

6  04  30 

2.38 

40 

580 

28  02  00 

23.58 

40 

280 

6  32  00 

2.65 

50 

590 

29  00  30 

24.82 

50 

290 

7  00  30 

2.95 

1000 

600 

30  00  00 

26.10 

5  00 

300 

7  30  00 

3.26 

30 

630 

33  04  30 

30.21 

10 

310 

8  00  30 

3.60 

11  00 

660 

36  18  00 

34.74 

20 

320 

8  32  00 

3.96 

30 

690 

39  40  30 

39.69 

30 

330 

9  04  30 

4.34 

12  00 

720 

43  12  00 

45.10 

40 

340 

9  38  00 

5.75 

30 

750 

46  52  30 

50.98 

50 

350 

10  12  30 

5.18 

13  00 

780 

50  42  00 

57.34 

600 

360 

10  48  00 

5.64 

14  00 

840 

58  48  00 

61.62 

10 

370 

11  24  30 

6.12 

15  00 

900 

67  30  00 

8809 

20 

380 

12  02  00 

6.63 

16  00 

960 

76  48  00 

106.91 

30 

390 

12  40  30 

7.17 

17  00 

1020 

86  42  00 

128.23 

40 

400 

13  20  00 

7.73 

18  00 

1080 

97  12  00 

152.22 

50 

410 

14  00  30 

8.33 

19  00 

1140 

108  18  00 

179.02 

7  00 

420 

14  42  00 

8.95 

2000 

1200 

120  00  00 

208.80 

APPENDIX. 


TABLE  II. 

DEFLECTIONS  OF  A  No.  1  SPIRAL  FOR  DIFFERENT  VALUES  OF  d. 
(Deflections  for  spiral  of  different  number  in  direct  proportion.) 


d 

rf2 

600 

d 

r/2 
600 

d 

<PL 

600 

d 

d* 
600 

0 

0.0 

50 

4.1 

100 

16.7 

150 

37.5 

1 

0.0 

51 

4.3 

101 

17.0 

151 

38.0 

2 

0.0 

52 

4.5 

102 

17.3 

152 

38.5 

3 

0.0 

53 

4.7 

103 

17.7 

153 

39.0 

4 

0.0 

54 

4.9 

104 

18.0 

154 

39.5 

5 

0.0 

55 

5.0 

105 

18.4 

155 

40.0 

6 

0.1 

56 

5.2 

106 

18.7 

156 

40.6 

7 

0.1 

57 

5.4 

107 

19.1 

157 

41.1 

8 

0.1 

58 

5.6 

108 

19.4 

158 

41.6 

9 

0.1 

59 

5.8 

109 

19.8 

159 

42.1 

10 

0.2 

60 

6.0 

110 

20.2 

160 

42.7 

11 

0.2 

61 

6.2 

111 

20.5 

161 

43.2 

12 

0.2 

62 

6.4 

112 

20.9 

162 

43.7 

13 

0.3 

63 

6.6 

113 

21.3 

163 

44.3 

14 

0.3 

64 

6.8 

114 

21.7 

164 

44.8 

15 

0.4 

65 

7.0 

115 

22.0 

165 

45.4 

16 

0.4 

66 

7.3 

116 

22.4 

166 

45.9 

17 

0.5 

67 

7.5 

117 

22.8 

167 

46.5 

18 

0.5 

68 

7.7 

118 

23.2 

168 

47.0 

19 

0.6 

69 

7.9 

119 

23.6 

169 

47.6 

20 

0.7 

70 

8.2 

120 

24.0 

170 

48.2 

21 

0.7 

71 

8.4 

121 

24.4 

171 

48.7 

22 

0.8 

72 

8.6 

122 

24.8 

172 

49.3 

23 

0.9 

73 

8.9 

123 

25.2 

173 

49.9 

24 

1.0 

74 

9.1 

124 

25.6 

174 

50.5 

25 

1.0 

75 

9.4 

125 

26.0 

175 

51.0 

26 

1.1 

76 

9.6 

126 

26.5 

176 

51.6 

27 

1.2 

77 

9.9 

127 

26.9 

177 

52.2 

28 

1.3 

78 

10.1 

128 

27.3 

178 

52.8 

29 

1.4 

79 

10.4 

129 

27.7 

179 

53.4 

30 

1.5 

80 

10.7 

130 

28.2 

180 

54.0 

31 

1.6 

81 

11.0 

131 

28.6 

181 

54.6 

32 

1.7 

82 

11.2 

132 

29.0 

182 

55.2 

33 

1.8 

83 

11.5 

133 

29.5 

183 

55.8 

34 

1.9 

84 

11.8 

134 

29.9 

184 

56.4 

35 

2.0 

85 

12.0 

135 

30.4 

185 

57.0 

36 

2.2 

86 

12.3 

136 

30.8 

186 

57.7 

37 

2.3 

87 

12.6 

137 

31.3 

187 

58.3 

38 

2.4 

88 

12.9 

138 

31.7 

188 

58.9 

39 

2.5 

89 

13.2 

139 

32.2 

189 

595 

40 

2.7 

90 

13.5 

140 

32.7 

190 

60.2 

41 

2.8 

91 

13.8 

141 

33.1 

191 

60.8 

42 

2.9 

92 

14.1 

142 

33.6 

192 

61.4 

43 

3.1 

93 

14.4 

143 

34.1 

193 

62.1 

44 

3.2 

94 

14.7 

144 

34.6 

194 

62.7 

45 

3.4 

95 

15.0 

145 

35.0 

195 

63.4 

46 

3.5 

96 

15.4 

146 

35.5 

196 

64.0 

47 

3.7 

97 

15.7 

147 

36.0 

197 

64.7 

48 

3.8 

98 

16.0 

148 

365 

198 

65.3 

49 

4.0 

99 

16.3 

149 

37.0 

199 

66.0 

APPENDIX. 


TABLE  II. -Continued. 


d 

d2 
600 

d 

.* 

600 

d 

</2 

600 

d 

rf2 

600 

200 

66.7 

240 

96.0 

280 

130.7 

320 

170.7 

201 

67.3 

241 

96.8 

281 

131.6 

321 

171.7 

202 

68.0 

242 

97.6 

282 

132.5 

322 

172.8 

203 

68.7 

243 

98.4 

283 

133.5 

323 

173.9 

204 

69.4 

244 

99.2 

284 

134.4 

324 

175.0 

205 

70.1 

245 

100.0 

285 

135.4 

325 

176.0 

206 

70.7 

246 

100.9 

286 

136.3 

326 

177.1 

207 

71.4 

247 

101.7 

287 

137.3 

327 

178.2 

208 

72.1 

248 

102.5 

288 

138.2 

328 

179.3 

209 

72.8 

249 

103.3 

289 

139.2 

329 

180.4 

210 

73.5 

250 

104.2 

290 

140.2 

880 

181.5 

211 

74.2 

251 

105.0 

291 

141.1 

331 

182.6 

212 

74.9 

252 

105.8 

292 

142.1 

332 

183.7 

213 

75.6 

253 

106.7 

293 

143.1 

333 

184.8 

214 

76.3 

254 

107.5 

294 

144.1 

334 

185.9 

215 

77.0 

255 

108.4 

295 

145.0 

335 

187.0 

216 

77.8 

256 

109.2 

296 

146.0 

336 

188.2 

217 

78.5 

257 

110.1 

297 

147.0 

337 

189.3 

218 

79.2 

258 

110.9 

298 

148.0 

338 

190.4 

219 

79.8 

259 

111.8 

299 

149.0 

339 

191.5 

220 

80.7 

260 

112.7 

300 

150.0 

340 

192.7 

221 

81.4 

261 

113.5 

301 

151.0 

341 

193.8 

222 

82.1 

262 

114.4 

302 

152.0 

342 

194.9 

223 

82.9 

263 

115.3 

303 

153.0 

343 

196.1 

224 

83.6 

264 

116.2 

304 

154.0 

344 

197.2 

225 

84.4 

265 

117.0 

305 

155.0 

345 

198.4 

226 

85.1 

266 

117.8 

306 

156.1 

346 

199.6 

227 

85.9 

267 

118.8 

307 

157.1 

347 

200.7 

228 

86.6 

268 

119.7 

308 

158.1 

348 

201.8 

229 

87.4 

269 

120.6 

309 

159.1 

349 

203.0 

230 

88.2 

270 

121.5 

310 

160.2 

350 

204.2 

231 

88.9 

271 

122.4 

311 

161.2 

351 

205.3 

232 

89.7 

272 

123.3 

312 

162.2 

352 

206.5 

233 

90.5 

273 

124.2 

313 

163.3 

353 

207.7 

234 

91.3 

274 

125.1 

314 

164.3 

354 

208.9 

235 

92.0 

275 

126.0 

315 

165.4 

355 

210.0 

236 

92.8 

276 

127.0 

316 

166.4 

356 

211.2 

237 

93.6 

277 

127.9 

317 

167.5 

357 

212.4 

238 

94.4 

278 

128.8 

318 

168.5 

358 

213.6 

239 

95.2 

279 

129.7 

319 

169.6 

359 

214.8 

(L 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 

AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


194; 


133 


YA  01415 


